Uniform 7polytope
In sevendimensional geometry, a 7polytope is a polytope contained by 6polytope facets. Each 5polytope ridge being shared by exactly two 6polytope facets.
A uniform 7polytope is one whose symmetry group is transitive on vertices and whose facets are uniform 6polytopes.
Regular 7polytopes
Regular 7polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with u {p,q,r,s,t} 6polytopes facets around each 4face.
There are exactly three such convex regular 7polytopes:
 {3,3,3,3,3,3}  7simplex
 {4,3,3,3,3,3}  7cube
 {3,3,3,3,3,4}  7orthoplex
There are no nonconvex regular 7polytopes.
Characteristics
The topology of any given 7polytope is defined by its Betti numbers and torsion coefficients.[1]
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.[1]
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.[1]
Uniform 7polytopes by fundamental Coxeter groups
Uniform 7polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the CoxeterDynkin diagrams:
#  Coxeter group  Regular and semiregular forms  Uniform count  

1  A_{7}  [3^{6}] 

71  
2  B_{7}  [4,3^{5}] 

127 + 32  
3  D_{7}  [3^{3,1,1}] 

95 (0 unique)  
4  E_{7}  [3^{3,2,1}]  127 
Prismatic finite Coxeter groups  

#  Coxeter group  Coxeter diagram  
6+1  
1  A_{6}A_{1}  [3^{5}]×[ ]  
2  BC_{6}A_{1}  [4,3^{4}]×[ ]  
3  D_{6}A_{1}  [3^{3,1,1}]×[ ]  
4  E_{6}A_{1}  [3^{2,2,1}]×[ ]  
5+2  
1  A_{5}I_{2}(p)  [3,3,3]×[p]  
2  BC_{5}I_{2}(p)  [4,3,3]×[p]  
3  D_{5}I_{2}(p)  [3^{2,1,1}]×[p]  
5+1+1  
1  A_{5}A_{1}^{2}  [3,3,3]×[ ]^{2}  
2  BC_{5}A_{1}^{2}  [4,3,3]×[ ]^{2}  
3  D_{5}A_{1}^{2}  [3^{2,1,1}]×[ ]^{2}  
4+3  
1  A_{4}A_{3}  [3,3,3]×[3,3]  
2  A_{4}B_{3}  [3,3,3]×[4,3]  
3  A_{4}H_{3}  [3,3,3]×[5,3]  
4  BC_{4}A_{3}  [4,3,3]×[3,3]  
5  BC_{4}B_{3}  [4,3,3]×[4,3]  
6  BC_{4}H_{3}  [4,3,3]×[5,3]  
7  H_{4}A_{3}  [5,3,3]×[3,3]  
8  H_{4}B_{3}  [5,3,3]×[4,3]  
9  H_{4}H_{3}  [5,3,3]×[5,3]  
10  F_{4}A_{3}  [3,4,3]×[3,3]  
11  F_{4}B_{3}  [3,4,3]×[4,3]  
12  F_{4}H_{3}  [3,4,3]×[5,3]  
13  D_{4}A_{3}  [3^{1,1,1}]×[3,3]  
14  D_{4}B_{3}  [3^{1,1,1}]×[4,3]  
15  D_{4}H_{3}  [3^{1,1,1}]×[5,3]  
4+2+1  
1  A_{4}I_{2}(p)A_{1}  [3,3,3]×[p]×[ ]  
2  BC_{4}I_{2}(p)A_{1}  [4,3,3]×[p]×[ ]  
3  F_{4}I_{2}(p)A_{1}  [3,4,3]×[p]×[ ]  
4  H_{4}I_{2}(p)A_{1}  [5,3,3]×[p]×[ ]  
5  D_{4}I_{2}(p)A_{1}  [3^{1,1,1}]×[p]×[ ]  
4+1+1+1  
1  A_{4}A_{1}^{3}  [3,3,3]×[ ]^{3}  
2  BC_{4}A_{1}^{3}  [4,3,3]×[ ]^{3}  
3  F_{4}A_{1}^{3}  [3,4,3]×[ ]^{3}  
4  H_{4}A_{1}^{3}  [5,3,3]×[ ]^{3}  
5  D_{4}A_{1}^{3}  [3^{1,1,1}]×[ ]^{3}  
3+3+1  
1  A_{3}A_{3}A_{1}  [3,3]×[3,3]×[ ]  
2  A_{3}B_{3}A_{1}  [3,3]×[4,3]×[ ]  
3  A_{3}H_{3}A_{1}  [3,3]×[5,3]×[ ]  
4  BC_{3}B_{3}A_{1}  [4,3]×[4,3]×[ ]  
5  BC_{3}H_{3}A_{1}  [4,3]×[5,3]×[ ]  
6  H_{3}A_{3}A_{1}  [5,3]×[5,3]×[ ]  
3+2+2  
1  A_{3}I_{2}(p)I_{2}(q)  [3,3]×[p]×[q]  
2  BC_{3}I_{2}(p)I_{2}(q)  [4,3]×[p]×[q]  
3  H_{3}I_{2}(p)I_{2}(q)  [5,3]×[p]×[q]  
3+2+1+1  
1  A_{3}I_{2}(p)A_{1}^{2}  [3,3]×[p]×[ ]^{2}  
2  BC_{3}I_{2}(p)A_{1}^{2}  [4,3]×[p]×[ ]^{2}  
3  H_{3}I_{2}(p)A_{1}^{2}  [5,3]×[p]×[ ]^{2}  
3+1+1+1+1  
1  A_{3}A_{1}^{4}  [3,3]×[ ]^{4}  
2  BC_{3}A_{1}^{4}  [4,3]×[ ]^{4}  
3  H_{3}A_{1}^{4}  [5,3]×[ ]^{4}  
2+2+2+1  
1  I_{2}(p)I_{2}(q)I_{2}(r)A_{1}  [p]×[q]×[r]×[ ]  
2+2+1+1+1  
1  I_{2}(p)I_{2}(q)A_{1}^{3}  [p]×[q]×[ ]^{3}  
2+1+1+1+1+1  
1  I_{2}(p)A_{1}^{5}  [p]×[ ]^{5}  
1+1+1+1+1+1+1  
1  A_{1}^{7}  [ ]^{7} 
The A_{7} family
The A_{7} family has symmetry of order 40320 (8 factorial).
There are 71 (64+81) forms based on all permutations of the CoxeterDynkin diagrams with one or more rings. All 71 are enumerated below. Norman Johnson's truncation names are given. Bowers names and acronym are also given for crossreferencing.
See also a list of A7 polytopes for symmetric Coxeter plane graphs of these polytopes.
A_{7} uniform polytopes  

#  CoxeterDynkin diagram  Truncation indices 
Johnson name Bowers name (and acronym) 
Basepoint  Element counts  
6  5  4  3  2  1  0  
1  t_{0}  7simplex (oca)  (0,0,0,0,0,0,0,1)  8  28  56  70  56  28  8  
2  t_{1}  Rectified 7simplex (roc)  (0,0,0,0,0,0,1,1)  16  84  224  350  336  168  28  
3  t_{2}  Birectified 7simplex (broc)  (0,0,0,0,0,1,1,1)  16  112  392  770  840  420  56  
4  t_{3}  Trirectified 7simplex (he)  (0,0,0,0,1,1,1,1)  16  112  448  980  1120  560  70  
5  t_{0,1}  Truncated 7simplex (toc)  (0,0,0,0,0,0,1,2)  16  84  224  350  336  196  56  
6  t_{0,2}  Cantellated 7simplex (saro)  (0,0,0,0,0,1,1,2)  44  308  980  1750  1876  1008  168  
7  t_{1,2}  Bitruncated 7simplex (bittoc)  (0,0,0,0,0,1,2,2)  588  168  
8  t_{0,3}  Runcinated 7simplex (spo)  (0,0,0,0,1,1,1,2)  100  756  2548  4830  4760  2100  280  
9  t_{1,3}  Bicantellated 7simplex (sabro)  (0,0,0,0,1,1,2,2)  2520  420  
10  t_{2,3}  Tritruncated 7simplex (tattoc)  (0,0,0,0,1,2,2,2)  980  280  
11  t_{0,4}  Stericated 7simplex (sco)  (0,0,0,1,1,1,1,2)  2240  280  
12  t_{1,4}  Biruncinated 7simplex (sibpo)  (0,0,0,1,1,1,2,2)  4200  560  
13  t_{2,4}  Tricantellated 7simplex (stiroh)  (0,0,0,1,1,2,2,2)  3360  560  
14  t_{0,5}  Pentellated 7simplex (seto)  (0,0,1,1,1,1,1,2)  1260  168  
15  t_{1,5}  Bistericated 7simplex (sabach)  (0,0,1,1,1,1,2,2)  3360  420  
16  t_{0,6}  Hexicated 7simplex (suph)  (0,1,1,1,1,1,1,2)  336  56  
17  t_{0,1,2}  Cantitruncated 7simplex (garo)  (0,0,0,0,0,1,2,3)  1176  336  
18  t_{0,1,3}  Runcitruncated 7simplex (patto)  (0,0,0,0,1,1,2,3)  4620  840  
19  t_{0,2,3}  Runcicantellated 7simplex (paro)  (0,0,0,0,1,2,2,3)  3360  840  
20  t_{1,2,3}  Bicantitruncated 7simplex (gabro)  (0,0,0,0,1,2,3,3)  2940  840  
21  t_{0,1,4}  Steritruncated 7simplex (cato)  (0,0,0,1,1,1,2,3)  7280  1120  
22  t_{0,2,4}  Stericantellated 7simplex (caro)  (0,0,0,1,1,2,2,3)  10080  1680  
23  t_{1,2,4}  Biruncitruncated 7simplex (bipto)  (0,0,0,1,1,2,3,3)  8400  1680  
24  t_{0,3,4}  Steriruncinated 7simplex (cepo)  (0,0,0,1,2,2,2,3)  5040  1120  
25  t_{1,3,4}  Biruncicantellated 7simplex (bipro)  (0,0,0,1,2,2,3,3)  7560  1680  
26  t_{2,3,4}  Tricantitruncated 7simplex (gatroh)  (0,0,0,1,2,3,3,3)  3920  1120  
27  t_{0,1,5}  Pentitruncated 7simplex (teto)  (0,0,1,1,1,1,2,3)  5460  840  
28  t_{0,2,5}  Penticantellated 7simplex (tero)  (0,0,1,1,1,2,2,3)  11760  1680  
29  t_{1,2,5}  Bisteritruncated 7simplex (bacto)  (0,0,1,1,1,2,3,3)  9240  1680  
30  t_{0,3,5}  Pentiruncinated 7simplex (tepo)  (0,0,1,1,2,2,2,3)  10920  1680  
31  t_{1,3,5}  Bistericantellated 7simplex (bacroh)  (0,0,1,1,2,2,3,3)  15120  2520  
32  t_{0,4,5}  Pentistericated 7simplex (teco)  (0,0,1,2,2,2,2,3)  4200  840  
33  t_{0,1,6}  Hexitruncated 7simplex (puto)  (0,1,1,1,1,1,2,3)  1848  336  
34  t_{0,2,6}  Hexicantellated 7simplex (puro)  (0,1,1,1,1,2,2,3)  5880  840  
35  t_{0,3,6}  Hexiruncinated 7simplex (puph)  (0,1,1,1,2,2,2,3)  8400  1120  
36  t_{0,1,2,3}  Runcicantitruncated 7simplex (gapo)  (0,0,0,0,1,2,3,4)  5880  1680  
37  t_{0,1,2,4}  Stericantitruncated 7simplex (cagro)  (0,0,0,1,1,2,3,4)  16800  3360  
38  t_{0,1,3,4}  Steriruncitruncated 7simplex (capto)  (0,0,0,1,2,2,3,4)  13440  3360  
39  t_{0,2,3,4}  Steriruncicantellated 7simplex (capro)  (0,0,0,1,2,3,3,4)  13440  3360  
40  t_{1,2,3,4}  Biruncicantitruncated 7simplex (gibpo)  (0,0,0,1,2,3,4,4)  11760  3360  
41  t_{0,1,2,5}  Penticantitruncated 7simplex (tegro)  (0,0,1,1,1,2,3,4)  18480  3360  
42  t_{0,1,3,5}  Pentiruncitruncated 7simplex (tapto)  (0,0,1,1,2,2,3,4)  27720  5040  
43  t_{0,2,3,5}  Pentiruncicantellated 7simplex (tapro)  (0,0,1,1,2,3,3,4)  25200  5040  
44  t_{1,2,3,5}  Bistericantitruncated 7simplex (bacogro)  (0,0,1,1,2,3,4,4)  22680  5040  
45  t_{0,1,4,5}  Pentisteritruncated 7simplex (tecto)  (0,0,1,2,2,2,3,4)  15120  3360  
46  t_{0,2,4,5}  Pentistericantellated 7simplex (tecro)  (0,0,1,2,2,3,3,4)  25200  5040  
47  t_{1,2,4,5}  Bisteriruncitruncated 7simplex (bicpath)  (0,0,1,2,2,3,4,4)  20160  5040  
48  t_{0,3,4,5}  Pentisteriruncinated 7simplex (tacpo)  (0,0,1,2,3,3,3,4)  15120  3360  
49  t_{0,1,2,6}  Hexicantitruncated 7simplex (pugro)  (0,1,1,1,1,2,3,4)  8400  1680  
50  t_{0,1,3,6}  Hexiruncitruncated 7simplex (pugato)  (0,1,1,1,2,2,3,4)  20160  3360  
51  t_{0,2,3,6}  Hexiruncicantellated 7simplex (pugro)  (0,1,1,1,2,3,3,4)  16800  3360  
52  t_{0,1,4,6}  Hexisteritruncated 7simplex (pucto)  (0,1,1,2,2,2,3,4)  20160  3360  
53  t_{0,2,4,6}  Hexistericantellated 7simplex (pucroh)  (0,1,1,2,2,3,3,4)  30240  5040  
54  t_{0,1,5,6}  Hexipentitruncated 7simplex (putath)  (0,1,2,2,2,2,3,4)  8400  1680  
55  t_{0,1,2,3,4}  Steriruncicantitruncated 7simplex (gecco)  (0,0,0,1,2,3,4,5)  23520  6720  
56  t_{0,1,2,3,5}  Pentiruncicantitruncated 7simplex (tegapo)  (0,0,1,1,2,3,4,5)  45360  10080  
57  t_{0,1,2,4,5}  Pentistericantitruncated 7simplex (tecagro)  (0,0,1,2,2,3,4,5)  40320  10080  
58  t_{0,1,3,4,5}  Pentisteriruncitruncated 7simplex (tacpeto)  (0,0,1,2,3,3,4,5)  40320  10080  
59  t_{0,2,3,4,5}  Pentisteriruncicantellated 7simplex (tacpro)  (0,0,1,2,3,4,4,5)  40320  10080  
60  t_{1,2,3,4,5}  Bisteriruncicantitruncated 7simplex (gabach)  (0,0,1,2,3,4,5,5)  35280  10080  
61  t_{0,1,2,3,6}  Hexiruncicantitruncated 7simplex (pugopo)  (0,1,1,1,2,3,4,5)  30240  6720  
62  t_{0,1,2,4,6}  Hexistericantitruncated 7simplex (pucagro)  (0,1,1,2,2,3,4,5)  50400  10080  
63  t_{0,1,3,4,6}  Hexisteriruncitruncated 7simplex (pucpato)  (0,1,1,2,3,3,4,5)  45360  10080  
64  t_{0,2,3,4,6}  Hexisteriruncicantellated 7simplex (pucproh)  (0,1,1,2,3,4,4,5)  45360  10080  
65  t_{0,1,2,5,6}  Hexipenticantitruncated 7simplex (putagro)  (0,1,2,2,2,3,4,5)  30240  6720  
66  t_{0,1,3,5,6}  Hexipentiruncitruncated 7simplex (putpath)  (0,1,2,2,3,3,4,5)  50400  10080  
67  t_{0,1,2,3,4,5}  Pentisteriruncicantitruncated 7simplex (geto)  (0,0,1,2,3,4,5,6)  70560  20160  
68  t_{0,1,2,3,4,6}  Hexisteriruncicantitruncated 7simplex (pugaco)  (0,1,1,2,3,4,5,6)  80640  20160  
69  t_{0,1,2,3,5,6}  Hexipentiruncicantitruncated 7simplex (putgapo)  (0,1,2,2,3,4,5,6)  80640  20160  
70  t_{0,1,2,4,5,6}  Hexipentistericantitruncated 7simplex (putcagroh)  (0,1,2,3,3,4,5,6)  80640  20160  
71  t_{0,1,2,3,4,5,6}  Omnitruncated 7simplex (guph)  (0,1,2,3,4,5,6,7)  141120  40320 
The B_{7} family
The B_{7} family has symmetry of order 645120 (7 factorial x 2^{7}).
There are 127 forms based on all permutations of the CoxeterDynkin diagrams with one or more rings. Johnson and Bowers names.
See also a list of B7 polytopes for symmetric Coxeter plane graphs of these polytopes.
B_{7} uniform polytopes  

#  CoxeterDynkin diagram tnotation 
Name (BSA)  Base point  Element counts  
6  5  4  3  2  1  0  
1  t_{0}{3,3,3,3,3,4}  7orthoplex (zee)  (0,0,0,0,0,0,1)√2  128  448  672  560  280  84  14  
2  t_{1}{3,3,3,3,3,4}  Rectified 7orthoplex (rez)  (0,0,0,0,0,1,1)√2  142  1344  3360  3920  2520  840  84  
3  t_{2}{3,3,3,3,3,4}  Birectified 7orthoplex (barz)  (0,0,0,0,1,1,1)√2  142  1428  6048  10640  8960  3360  280  
4  t_{3}{4,3,3,3,3,3}  Trirectified 7cube (sez)  (0,0,0,1,1,1,1)√2  142  1428  6328  14560  15680  6720  560  
5  t_{2}{4,3,3,3,3,3}  Birectified 7cube (bersa)  (0,0,1,1,1,1,1)√2  142  1428  5656  11760  13440  6720  672  
6  t_{1}{4,3,3,3,3,3}  Rectified 7cube (rasa)  (0,1,1,1,1,1,1)√2  142  980  2968  5040  5152  2688  448  
7  t_{0}{4,3,3,3,3,3}  7cube (hept)  (0,0,0,0,0,0,0)√2 + (1,1,1,1,1,1,1)  14  84  280  560  672  448  128  
8  t_{0,1}{3,3,3,3,3,4}  Truncated 7orthoplex (Taz)  (0,0,0,0,0,1,2)√2  142  1344  3360  4760  2520  924  168  
9  t_{0,2}{3,3,3,3,3,4}  Cantellated 7orthoplex (Sarz)  (0,0,0,0,1,1,2)√2  226  4200  15456  24080  19320  7560  840  
10  t_{1,2}{3,3,3,3,3,4}  Bitruncated 7orthoplex (Botaz)  (0,0,0,0,1,2,2)√2  4200  840  
11  t_{0,3}{3,3,3,3,3,4}  Runcinated 7orthoplex (Spaz)  (0,0,0,1,1,1,2)√2  23520  2240  
12  t_{1,3}{3,3,3,3,3,4}  Bicantellated 7orthoplex (Sebraz)  (0,0,0,1,1,2,2)√2  26880  3360  
13  t_{2,3}{3,3,3,3,3,4}  Tritruncated 7orthoplex (Totaz)  (0,0,0,1,2,2,2)√2  10080  2240  
14  t_{0,4}{3,3,3,3,3,4}  Stericated 7orthoplex (Scaz)  (0,0,1,1,1,1,2)√2  33600  3360  
15  t_{1,4}{3,3,3,3,3,4}  Biruncinated 7orthoplex (Sibpaz)  (0,0,1,1,1,2,2)√2  60480  6720  
16  t_{2,4}{4,3,3,3,3,3}  Tricantellated 7cube (Strasaz)  (0,0,1,1,2,2,2)√2  47040  6720  
17  t_{2,3}{4,3,3,3,3,3}  Tritruncated 7cube (Tatsa)  (0,0,1,2,2,2,2)√2  13440  3360  
18  t_{0,5}{3,3,3,3,3,4}  Pentellated 7orthoplex (Staz)  (0,1,1,1,1,1,2)√2  20160  2688  
19  t_{1,5}{4,3,3,3,3,3}  Bistericated 7cube (Sabcosaz)  (0,1,1,1,1,2,2)√2  53760  6720  
20  t_{1,4}{4,3,3,3,3,3}  Biruncinated 7cube (Sibposa)  (0,1,1,1,2,2,2)√2  67200  8960  
21  t_{1,3}{4,3,3,3,3,3}  Bicantellated 7cube (Sibrosa)  (0,1,1,2,2,2,2)√2  40320  6720  
22  t_{1,2}{4,3,3,3,3,3}  Bitruncated 7cube (Betsa)  (0,1,2,2,2,2,2)√2  9408  2688  
23  t_{0,6}{4,3,3,3,3,3}  Hexicated 7cube (Supposaz)  (0,0,0,0,0,0,1)√2 + (1,1,1,1,1,1,1)  5376  896  
24  t_{0,5}{4,3,3,3,3,3}  Pentellated 7cube (Stesa)  (0,0,0,0,0,1,1)√2 + (1,1,1,1,1,1,1)  20160  2688  
25  t_{0,4}{4,3,3,3,3,3}  Stericated 7cube (Scosa)  (0,0,0,0,1,1,1)√2 + (1,1,1,1,1,1,1)  35840  4480  
26  t_{0,3}{4,3,3,3,3,3}  Runcinated 7cube (Spesa)  (0,0,0,1,1,1,1)√2 + (1,1,1,1,1,1,1)  33600  4480  
27  t_{0,2}{4,3,3,3,3,3}  Cantellated 7cube (Sersa)  (0,0,1,1,1,1,1)√2 + (1,1,1,1,1,1,1)  16128  2688  
28  t_{0,1}{4,3,3,3,3,3}  Truncated 7cube (Tasa)  (0,1,1,1,1,1,1)√2 + (1,1,1,1,1,1,1)  142  980  2968  5040  5152  3136  896  
29  t_{0,1,2}{3,3,3,3,3,4}  Cantitruncated 7orthoplex (Garz)  (0,1,2,3,3,3,3)√2  8400  1680  
30  t_{0,1,3}{3,3,3,3,3,4}  Runcitruncated 7orthoplex (Potaz)  (0,1,2,2,3,3,3)√2  50400  6720  
31  t_{0,2,3}{3,3,3,3,3,4}  Runcicantellated 7orthoplex (Parz)  (0,1,1,2,3,3,3)√2  33600  6720  
32  t_{1,2,3}{3,3,3,3,3,4}  Bicantitruncated 7orthoplex (Gebraz)  (0,0,1,2,3,3,3)√2  30240  6720  
33  t_{0,1,4}{3,3,3,3,3,4}  Steritruncated 7orthoplex (Catz)  (0,0,1,1,1,2,3)√2  107520  13440  
34  t_{0,2,4}{3,3,3,3,3,4}  Stericantellated 7orthoplex (Craze)  (0,0,1,1,2,2,3)√2  141120  20160  
35  t_{1,2,4}{3,3,3,3,3,4}  Biruncitruncated 7orthoplex (Baptize)  (0,0,1,1,2,3,3)√2  120960  20160  
36  t_{0,3,4}{3,3,3,3,3,4}  Steriruncinated 7orthoplex (Copaz)  (0,1,1,1,2,3,3)√2  67200  13440  
37  t_{1,3,4}{3,3,3,3,3,4}  Biruncicantellated 7orthoplex (Boparz)  (0,0,1,2,2,3,3)√2  100800  20160  
38  t_{2,3,4}{4,3,3,3,3,3}  Tricantitruncated 7cube (Gotrasaz)  (0,0,0,1,2,3,3)√2  53760  13440  
39  t_{0,1,5}{3,3,3,3,3,4}  Pentitruncated 7orthoplex (Tetaz)  (0,1,1,1,1,2,3)√2  87360  13440  
40  t_{0,2,5}{3,3,3,3,3,4}  Penticantellated 7orthoplex (Teroz)  (0,1,1,1,2,2,3)√2  188160  26880  
41  t_{1,2,5}{3,3,3,3,3,4}  Bisteritruncated 7orthoplex (Boctaz)  (0,1,1,1,2,3,3)√2  147840  26880  
42  t_{0,3,5}{3,3,3,3,3,4}  Pentiruncinated 7orthoplex (Topaz)  (0,1,1,2,2,2,3)√2  174720  26880  
43  t_{1,3,5}{4,3,3,3,3,3}  Bistericantellated 7cube (Bacresaz)  (0,1,1,2,2,3,3)√2  241920  40320  
44  t_{1,3,4}{4,3,3,3,3,3}  Biruncicantellated 7cube (Bopresa)  (0,1,1,2,3,3,3)√2  120960  26880  
45  t_{0,4,5}{3,3,3,3,3,4}  Pentistericated 7orthoplex (Tocaz)  (0,1,2,2,2,2,3)√2  67200  13440  
46  t_{1,2,5}{4,3,3,3,3,3}  Bisteritruncated 7cube (Bactasa)  (0,1,2,2,2,3,3)√2  147840  26880  
47  t_{1,2,4}{4,3,3,3,3,3}  Biruncitruncated 7cube (Biptesa)  (0,1,2,2,3,3,3)√2  134400  26880  
48  t_{1,2,3}{4,3,3,3,3,3}  Bicantitruncated 7cube (Gibrosa)  (0,1,2,3,3,3,3)√2  47040  13440  
49  t_{0,1,6}{3,3,3,3,3,4}  Hexitruncated 7orthoplex (Putaz)  (0,0,0,0,0,1,2)√2 + (1,1,1,1,1,1,1)  29568  5376  
50  t_{0,2,6}{3,3,3,3,3,4}  Hexicantellated 7orthoplex (Puraz)  (0,0,0,0,1,1,2)√2 + (1,1,1,1,1,1,1)  94080  13440  
51  t_{0,4,5}{4,3,3,3,3,3}  Pentistericated 7cube (Tacosa)  (0,0,0,0,1,2,2)√2 + (1,1,1,1,1,1,1)  67200  13440  
52  t_{0,3,6}{4,3,3,3,3,3}  Hexiruncinated 7cube (Pupsez)  (0,0,0,1,1,1,2)√2 + (1,1,1,1,1,1,1)  134400  17920  
53  t_{0,3,5}{4,3,3,3,3,3}  Pentiruncinated 7cube (Tapsa)  (0,0,0,1,1,2,2)√2 + (1,1,1,1,1,1,1)  174720  26880  
54  t_{0,3,4}{4,3,3,3,3,3}  Steriruncinated 7cube (Capsa)  (0,0,0,1,2,2,2)√2 + (1,1,1,1,1,1,1)  80640  17920  
55  t_{0,2,6}{4,3,3,3,3,3}  Hexicantellated 7cube (Purosa)  (0,0,1,1,1,1,2)√2 + (1,1,1,1,1,1,1)  94080  13440  
56  t_{0,2,5}{4,3,3,3,3,3}  Penticantellated 7cube (Tersa)  (0,0,1,1,1,2,2)√2 + (1,1,1,1,1,1,1)  188160  26880  
57  t_{0,2,4}{4,3,3,3,3,3}  Stericantellated 7cube (Carsa)  (0,0,1,1,2,2,2)√2 + (1,1,1,1,1,1,1)  161280  26880  
58  t_{0,2,3}{4,3,3,3,3,3}  Runcicantellated 7cube (Parsa)  (0,0,1,2,2,2,2)√2 + (1,1,1,1,1,1,1)  53760  13440  
59  t_{0,1,6}{4,3,3,3,3,3}  Hexitruncated 7cube (Putsa)  (0,1,1,1,1,1,2)√2 + (1,1,1,1,1,1,1)  29568  5376  
60  t_{0,1,5}{4,3,3,3,3,3}  Pentitruncated 7cube (Tetsa)  (0,1,1,1,1,2,2)√2 + (1,1,1,1,1,1,1)  87360  13440  
61  t_{0,1,4}{4,3,3,3,3,3}  Steritruncated 7cube (Catsa)  (0,1,1,1,2,2,2)√2 + (1,1,1,1,1,1,1)  116480  17920  
62  t_{0,1,3}{4,3,3,3,3,3}  Runcitruncated 7cube (Petsa)  (0,1,1,2,2,2,2)√2 + (1,1,1,1,1,1,1)  73920  13440  
63  t_{0,1,2}{4,3,3,3,3,3}  Cantitruncated 7cube (Gersa)  (0,1,2,2,2,2,2)√2 + (1,1,1,1,1,1,1)  18816  5376  
64  t_{0,1,2,3}{3,3,3,3,3,4}  Runcicantitruncated 7orthoplex (Gopaz)  (0,1,2,3,4,4,4)√2  60480  13440  
65  t_{0,1,2,4}{3,3,3,3,3,4}  Stericantitruncated 7orthoplex (Cogarz)  (0,0,1,1,2,3,4)√2  241920  40320  
66  t_{0,1,3,4}{3,3,3,3,3,4}  Steriruncitruncated 7orthoplex (Captaz)  (0,0,1,2,2,3,4)√2  181440  40320  
67  t_{0,2,3,4}{3,3,3,3,3,4}  Steriruncicantellated 7orthoplex (Caparz)  (0,0,1,2,3,3,4)√2  181440  40320  
68  t_{1,2,3,4}{3,3,3,3,3,4}  Biruncicantitruncated 7orthoplex (Gibpaz)  (0,0,1,2,3,4,4)√2  161280  40320  
69  t_{0,1,2,5}{3,3,3,3,3,4}  Penticantitruncated 7orthoplex (Tograz)  (0,1,1,1,2,3,4)√2  295680  53760  
70  t_{0,1,3,5}{3,3,3,3,3,4}  Pentiruncitruncated 7orthoplex (Toptaz)  (0,1,1,2,2,3,4)√2  443520  80640  
71  t_{0,2,3,5}{3,3,3,3,3,4}  Pentiruncicantellated 7orthoplex (Toparz)  (0,1,1,2,3,3,4)√2  403200  80640  
72  t_{1,2,3,5}{3,3,3,3,3,4}  Bistericantitruncated 7orthoplex (Becogarz)  (0,1,1,2,3,4,4)√2  362880  80640  
73  t_{0,1,4,5}{3,3,3,3,3,4}  Pentisteritruncated 7orthoplex (Tacotaz)  (0,1,2,2,2,3,4)√2  241920  53760  
74  t_{0,2,4,5}{3,3,3,3,3,4}  Pentistericantellated 7orthoplex (Tocarz)  (0,1,2,2,3,3,4)√2  403200  80640  
75  t_{1,2,4,5}{4,3,3,3,3,3}  Bisteriruncitruncated 7cube (Bocaptosaz)  (0,1,2,2,3,4,4)√2  322560  80640  
76  t_{0,3,4,5}{3,3,3,3,3,4}  Pentisteriruncinated 7orthoplex (Tecpaz)  (0,1,2,3,3,3,4)√2  241920  53760  
77  t_{1,2,3,5}{4,3,3,3,3,3}  Bistericantitruncated 7cube (Becgresa)  (0,1,2,3,3,4,4)√2  362880  80640  
78  t_{1,2,3,4}{4,3,3,3,3,3}  Biruncicantitruncated 7cube (Gibposa)  (0,1,2,3,4,4,4)√2  188160  53760  
79  t_{0,1,2,6}{3,3,3,3,3,4}  Hexicantitruncated 7orthoplex (Pugarez)  (0,0,0,0,1,2,3)√2 + (1,1,1,1,1,1,1)  134400  26880  
80  t_{0,1,3,6}{3,3,3,3,3,4}  Hexiruncitruncated 7orthoplex (Papataz)  (0,0,0,1,1,2,3)√2 + (1,1,1,1,1,1,1)  322560  53760  
81  t_{0,2,3,6}{3,3,3,3,3,4}  Hexiruncicantellated 7orthoplex (Puparez)  (0,0,0,1,2,2,3)√2 + (1,1,1,1,1,1,1)  268800  53760  
82  t_{0,3,4,5}{4,3,3,3,3,3}  Pentisteriruncinated 7cube (Tecpasa)  (0,0,0,1,2,3,3)√2 + (1,1,1,1,1,1,1)  241920  53760  
83  t_{0,1,4,6}{3,3,3,3,3,4}  Hexisteritruncated 7orthoplex (Pucotaz)  (0,0,1,1,1,2,3)√2 + (1,1,1,1,1,1,1)  322560  53760  
84  t_{0,2,4,6}{4,3,3,3,3,3}  Hexistericantellated 7cube (Pucrosaz)  (0,0,1,1,2,2,3)√2 + (1,1,1,1,1,1,1)  483840  80640  
85  t_{0,2,4,5}{4,3,3,3,3,3}  Pentistericantellated 7cube (Tecresa)  (0,0,1,1,2,3,3)√2 + (1,1,1,1,1,1,1)  403200  80640  
86  t_{0,2,3,6}{4,3,3,3,3,3}  Hexiruncicantellated 7cube (Pupresa)  (0,0,1,2,2,2,3)√2 + (1,1,1,1,1,1,1)  268800  53760  
87  t_{0,2,3,5}{4,3,3,3,3,3}  Pentiruncicantellated 7cube (Topresa)  (0,0,1,2,2,3,3)√2 + (1,1,1,1,1,1,1)  403200  80640  
88  t_{0,2,3,4}{4,3,3,3,3,3}  Steriruncicantellated 7cube (Copresa)  (0,0,1,2,3,3,3)√2 + (1,1,1,1,1,1,1)  215040  53760  
89  t_{0,1,5,6}{4,3,3,3,3,3}  Hexipentitruncated 7cube (Putatosez)  (0,1,1,1,1,2,3)√2 + (1,1,1,1,1,1,1)  134400  26880  
90  t_{0,1,4,6}{4,3,3,3,3,3}  Hexisteritruncated 7cube (Pacutsa)  (0,1,1,1,2,2,3)√2 + (1,1,1,1,1,1,1)  322560  53760  
91  t_{0,1,4,5}{4,3,3,3,3,3}  Pentisteritruncated 7cube (Tecatsa)  (0,1,1,1,2,3,3)√2 + (1,1,1,1,1,1,1)  241920  53760  
92  t_{0,1,3,6}{4,3,3,3,3,3}  Hexiruncitruncated 7cube (Pupetsa)  (0,1,1,2,2,2,3)√2 + (1,1,1,1,1,1,1)  322560  53760  
93  t_{0,1,3,5}{4,3,3,3,3,3}  Pentiruncitruncated 7cube (Toptosa)  (0,1,1,2,2,3,3)√2 + (1,1,1,1,1,1,1)  443520  80640  
94  t_{0,1,3,4}{4,3,3,3,3,3}  Steriruncitruncated 7cube (Captesa)  (0,1,1,2,3,3,3)√2 + (1,1,1,1,1,1,1)  215040  53760  
95  t_{0,1,2,6}{4,3,3,3,3,3}  Hexicantitruncated 7cube (Pugrosa)  (0,1,2,2,2,2,3)√2 + (1,1,1,1,1,1,1)  134400  26880  
96  t_{0,1,2,5}{4,3,3,3,3,3}  Penticantitruncated 7cube (Togresa)  (0,1,2,2,2,3,3)√2 + (1,1,1,1,1,1,1)  295680  53760  
97  t_{0,1,2,4}{4,3,3,3,3,3}  Stericantitruncated 7cube (Cogarsa)  (0,1,2,2,3,3,3)√2 + (1,1,1,1,1,1,1)  268800  53760  
98  t_{0,1,2,3}{4,3,3,3,3,3}  Runcicantitruncated 7cube (Gapsa)  (0,1,2,3,3,3,3)√2 + (1,1,1,1,1,1,1)  94080  26880  
99  t_{0,1,2,3,4}{3,3,3,3,3,4}  Steriruncicantitruncated 7orthoplex (Gocaz)  (0,0,1,2,3,4,5)√2  322560  80640  
100  t_{0,1,2,3,5}{3,3,3,3,3,4}  Pentiruncicantitruncated 7orthoplex (Tegopaz)  (0,1,1,2,3,4,5)√2  725760  161280  
101  t_{0,1,2,4,5}{3,3,3,3,3,4}  Pentistericantitruncated 7orthoplex (Tecagraz)  (0,1,2,2,3,4,5)√2  645120  161280  
102  t_{0,1,3,4,5}{3,3,3,3,3,4}  Pentisteriruncitruncated 7orthoplex (Tecpotaz)  (0,1,2,3,3,4,5)√2  645120  161280  
103  t_{0,2,3,4,5}{3,3,3,3,3,4}  Pentisteriruncicantellated 7orthoplex (Tacparez)  (0,1,2,3,4,4,5)√2  645120  161280  
104  t_{1,2,3,4,5}{4,3,3,3,3,3}  Bisteriruncicantitruncated 7cube (Gabcosaz)  (0,1,2,3,4,5,5)√2  564480  161280  
105  t_{0,1,2,3,6}{3,3,3,3,3,4}  Hexiruncicantitruncated 7orthoplex (Pugopaz)  (0,0,0,1,2,3,4)√2 + (1,1,1,1,1,1,1)  483840  107520  
106  t_{0,1,2,4,6}{3,3,3,3,3,4}  Hexistericantitruncated 7orthoplex (Pucagraz)  (0,0,1,1,2,3,4)√2 + (1,1,1,1,1,1,1)  806400  161280  
107  t_{0,1,3,4,6}{3,3,3,3,3,4}  Hexisteriruncitruncated 7orthoplex (Pucpotaz)  (0,0,1,2,2,3,4)√2 + (1,1,1,1,1,1,1)  725760  161280  
108  t_{0,2,3,4,6}{4,3,3,3,3,3}  Hexisteriruncicantellated 7cube (Pucprosaz)  (0,0,1,2,3,3,4)√2 + (1,1,1,1,1,1,1)  725760  161280  
109  t_{0,2,3,4,5}{4,3,3,3,3,3}  Pentisteriruncicantellated 7cube (Tocpresa)  (0,0,1,2,3,4,4)√2 + (1,1,1,1,1,1,1)  645120  161280  
110  t_{0,1,2,5,6}{3,3,3,3,3,4}  Hexipenticantitruncated 7orthoplex (Putegraz)  (0,1,1,1,2,3,4)√2 + (1,1,1,1,1,1,1)  483840  107520  
111  t_{0,1,3,5,6}{4,3,3,3,3,3}  Hexipentiruncitruncated 7cube (Putpetsaz)  (0,1,1,2,2,3,4)√2 + (1,1,1,1,1,1,1)  806400  161280  
112  t_{0,1,3,4,6}{4,3,3,3,3,3}  Hexisteriruncitruncated 7cube (Pucpetsa)  (0,1,1,2,3,3,4)√2 + (1,1,1,1,1,1,1)  725760  161280  
113  t_{0,1,3,4,5}{4,3,3,3,3,3}  Pentisteriruncitruncated 7cube (Tecpetsa)  (0,1,1,2,3,4,4)√2 + (1,1,1,1,1,1,1)  645120  161280  
114  t_{0,1,2,5,6}{4,3,3,3,3,3}  Hexipenticantitruncated 7cube (Putgresa)  (0,1,2,2,2,3,4)√2 + (1,1,1,1,1,1,1)  483840  107520  
115  t_{0,1,2,4,6}{4,3,3,3,3,3}  Hexistericantitruncated 7cube (Pucagrosa)  (0,1,2,2,3,3,4)√2 + (1,1,1,1,1,1,1)  806400  161280  
116  t_{0,1,2,4,5}{4,3,3,3,3,3}  Pentistericantitruncated 7cube (Tecgresa)  (0,1,2,2,3,4,4)√2 + (1,1,1,1,1,1,1)  645120  161280  
117  t_{0,1,2,3,6}{4,3,3,3,3,3}  Hexiruncicantitruncated 7cube (Pugopsa)  (0,1,2,3,3,3,4)√2 + (1,1,1,1,1,1,1)  483840  107520  
118  t_{0,1,2,3,5}{4,3,3,3,3,3}  Pentiruncicantitruncated 7cube (Togapsa)  (0,1,2,3,3,4,4)√2 + (1,1,1,1,1,1,1)  725760  161280  
119  t_{0,1,2,3,4}{4,3,3,3,3,3}  Steriruncicantitruncated 7cube (Gacosa)  (0,1,2,3,4,4,4)√2 + (1,1,1,1,1,1,1)  376320  107520  
120  t_{0,1,2,3,4,5}{3,3,3,3,3,4}  Pentisteriruncicantitruncated 7orthoplex (Gotaz)  (0,1,2,3,4,5,6)√2  1128960  322560  
121  t_{0,1,2,3,4,6}{3,3,3,3,3,4}  Hexisteriruncicantitruncated 7orthoplex (Pugacaz)  (0,0,1,2,3,4,5)√2 + (1,1,1,1,1,1,1)  1290240  322560  
122  t_{0,1,2,3,5,6}{3,3,3,3,3,4}  Hexipentiruncicantitruncated 7orthoplex (Putgapaz)  (0,1,1,2,3,4,5)√2 + (1,1,1,1,1,1,1)  1290240  322560  
123  t_{0,1,2,4,5,6}{4,3,3,3,3,3}  Hexipentistericantitruncated 7cube (Putcagrasaz)  (0,1,2,2,3,4,5)√2 + (1,1,1,1,1,1,1)  1290240  322560  
124  t_{0,1,2,3,5,6}{4,3,3,3,3,3}  Hexipentiruncicantitruncated 7cube (Putgapsa)  (0,1,2,3,3,4,5)√2 + (1,1,1,1,1,1,1)  1290240  322560  
125  t_{0,1,2,3,4,6}{4,3,3,3,3,3}  Hexisteriruncicantitruncated 7cube (Pugacasa)  (0,1,2,3,4,4,5)√2 + (1,1,1,1,1,1,1)  1290240  322560  
126  t_{0,1,2,3,4,5}{4,3,3,3,3,3}  Pentisteriruncicantitruncated 7cube (Gotesa)  (0,1,2,3,4,5,5)√2 + (1,1,1,1,1,1,1)  1128960  322560  
127  t_{0,1,2,3,4,5,6}{4,3,3,3,3,3}  Omnitruncated 7cube (Guposaz)  (0,1,2,3,4,5,6)√2 + (1,1,1,1,1,1,1)  2257920  645120 
The D_{7} family
The D_{7} family has symmetry of order 322560 (7 factorial x 2^{6}).
This family has 3×32−1=95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D_{7} CoxeterDynkin diagram. Of these, 63 (2×32−1) are repeated from the B_{7} family and 32 are unique to this family, listed below. Bowers names and acronym are given for crossreferencing.
See also list of D7 polytopes for Coxeter plane graphs of these polytopes.
D_{7} uniform polytopes  

#  Coxeter diagram  Names  Base point (Alternately signed) 
Element counts  
6  5  4  3  2  1  0  
1  7cube demihepteract (hesa)  (1,1,1,1,1,1,1)  78  532  1624  2800  2240  672  64  
2  cantic 7cube truncated demihepteract (thesa)  (1,1,3,3,3,3,3)  142  1428  5656  11760  13440  7392  1344  
3  runcic 7cube small rhombated demihepteract (sirhesa)  (1,1,1,3,3,3,3)  16800  2240  
4  steric 7cube small prismated demihepteract (sphosa)  (1,1,1,1,3,3,3)  20160  2240  
5  pentic 7cube small cellated demihepteract (sochesa)  (1,1,1,1,1,3,3)  13440  1344  
6  hexic 7cube small terated demihepteract (suthesa)  (1,1,1,1,1,1,3)  4704  448  
7  runcicantic 7cube great rhombated demihepteract (Girhesa)  (1,1,3,5,5,5,5)  23520  6720  
8  stericantic 7cube prismatotruncated demihepteract (pothesa)  (1,1,3,3,5,5,5)  73920  13440  
9  steriruncic 7cube prismatorhomated demihepteract (prohesa)  (1,1,1,3,5,5,5)  40320  8960  
10  penticantic 7cube cellitruncated demihepteract (cothesa)  (1,1,3,3,3,5,5)  87360  13440  
11  pentiruncic 7cube cellirhombated demihepteract (crohesa)  (1,1,1,3,3,5,5)  87360  13440  
12  pentisteric 7cube celliprismated demihepteract (caphesa)  (1,1,1,1,3,5,5)  40320  6720  
13  hexicantic 7cube tericantic demihepteract (tuthesa)  (1,1,3,3,3,3,5)  43680  6720  
14  hexiruncic 7cube terirhombated demihepteract (turhesa)  (1,1,1,3,3,3,5)  67200  8960  
15  hexisteric 7cube teriprismated demihepteract (tuphesa)  (1,1,1,1,3,3,5)  53760  6720  
16  hexipentic 7cube tericellated demihepteract (tuchesa)  (1,1,1,1,1,3,5)  21504  2688  
17  steriruncicantic 7cube great prismated demihepteract (Gephosa)  (1,1,3,5,7,7,7)  94080  26880  
18  pentiruncicantic 7cube celligreatorhombated demihepteract (cagrohesa)  (1,1,3,5,5,7,7)  181440  40320  
19  pentistericantic 7cube celliprismatotruncated demihepteract (capthesa)  (1,1,3,3,5,7,7)  181440  40320  
20  pentisteriruncic 7cube celliprismatorhombated demihepteract (coprahesa)  (1,1,1,3,5,7,7)  120960  26880  
21  hexiruncicantic 7cube terigreatorhombated demihepteract (tugrohesa)  (1,1,3,5,5,5,7)  120960  26880  
22  hexistericantic 7cube teriprismatotruncated demihepteract (tupthesa)  (1,1,3,3,5,5,7)  221760  40320  
23  hexisteriruncic 7cube teriprismatorhombated demihepteract (tuprohesa)  (1,1,1,3,5,5,7)  134400  26880  
24  hexipenticantic 7cube teriCellitruncated demihepteract (tucothesa)  (1,1,3,3,3,5,7)  147840  26880  
25  hexipentiruncic 7cube tericellirhombated demihepteract (tucrohesa)  (1,1,1,3,3,5,7)  161280  26880  
26  hexipentisteric 7cube tericelliprismated demihepteract (tucophesa)  (1,1,1,1,3,5,7)  80640  13440  
27  pentisteriruncicantic 7cube great cellated demihepteract (gochesa)  (1,1,3,5,7,9,9)  282240  80640  
28  hexisteriruncicantic 7cube terigreatoprimated demihepteract (tugphesa)  (1,1,3,5,7,7,9)  322560  80640  
29  hexipentiruncicantic 7cube tericelligreatorhombated demihepteract (tucagrohesa)  (1,1,3,5,5,7,9)  322560  80640  
30  hexipentistericantic 7cube tericelliprismatotruncated demihepteract (tucpathesa)  (1,1,3,3,5,7,9)  362880  80640  
31  hexipentisteriruncic 7cube tericellprismatorhombated demihepteract (tucprohesa)  (1,1,1,3,5,7,9)  241920  53760  
32  hexipentisteriruncicantic 7cube great terated demihepteract (guthesa)  (1,1,3,5,7,9,11)  564480  161280 
The E_{7} family
The E_{7} Coxeter group has order 2,903,040.
There are 127 forms based on all permutations of the CoxeterDynkin diagrams with one or more rings.
See also a list of E7 polytopes for symmetric Coxeter plane graphs of these polytopes.
E_{7} uniform polytopes  

#  CoxeterDynkin diagram Schläfli symbol 
Names  Element counts  
6  5  4  3  2  1  0  
1  2_{31} (laq)  632  4788  16128  20160  10080  2016  126  
2  Rectified 2_{31} (rolaq)  758  10332  47880  100800  90720  30240  2016  
3  Rectified 1_{32} (rolin)  758  12348  72072  191520  241920  120960  10080  
4  1_{32} (lin)  182  4284  23688  50400  40320  10080  576  
5  Birectified 3_{21} (branq)  758  12348  68040  161280  161280  60480  4032  
6  Rectified 3_{21} (ranq)  758  44352  70560  48384  11592  12096  756  
7  3_{21} (naq)  702  6048  12096  10080  4032  756  56  
8  Truncated 2_{31} (talq)  758  10332  47880  100800  90720  32256  4032  
9  Cantellated 2_{31} (sirlaq)  131040  20160  
10  Bitruncated 2_{31} (botlaq)  30240  
11  small demified 2_{31} (shilq)  2774  22428  78120  151200  131040  42336  4032  
12  demirectified 2_{31} (hirlaq)  12096  
13  truncated 1_{32} (tolin)  20160  
14  small demiprismated 2_{31} (shiplaq)  20160  
15  birectified 1_{32} (berlin)  758  22428  142632  403200  544320  302400  40320  
16  tritruncated 3_{21} (totanq)  40320  
17  demibirectified 3_{21} (hobranq)  20160  
18  small cellated 2_{31} (scalq)  7560  
19  small biprismated 2_{31} (sobpalq)  30240  
20  small birhombated 3_{21} (sabranq)  60480  
21  demirectified 3_{21} (harnaq)  12096  
22  bitruncated 3_{21} (botnaq)  12096  
23  small terated 3_{21} (stanq)  1512  
24  small demicellated 3_{21} (shocanq)  12096  
25  small prismated 3_{21} (spanq)  40320  
26  small demified 3_{21} (shanq)  4032  
27  small rhombated 3_{21} (sranq)  12096  
28  Truncated 3_{21} (tanq)  758  11592  48384  70560  44352  12852  1512  
29  great rhombated 2_{31} (girlaq)  60480  
30  demitruncated 2_{31} (hotlaq)  24192  
31  small demirhombated 2_{31} (sherlaq)  60480  
32  demibitruncated 2_{31} (hobtalq)  60480  
33  demiprismated 2_{31} (hiptalq)  80640  
34  demiprismatorhombated 2_{31} (hiprolaq)  120960  
35  bitruncated 1_{32} (batlin)  120960  
36  small prismated 2_{31} (spalq)  80640  
37  small rhombated 1_{32} (sirlin)  120960  
38  tritruncated 2_{31} (tatilq)  80640  
39  cellitruncated 2_{31} (catalaq)  60480  
40  cellirhombated 2_{31} (crilq)  362880  
41  biprismatotruncated 2_{31} (biptalq)  181440  
42  small prismated 1_{32} (seplin)  60480  
43  small biprismated 3_{21} (sabipnaq)  120960  
44  small demibirhombated 3_{21} (shobranq)  120960  
45  cellidemiprismated 2_{31} (chaplaq)  60480  
46  demibiprismatotruncated 3_{21} (hobpotanq)  120960  
47  great birhombated 3_{21} (gobranq)  120960  
48  demibitruncated 3_{21} (hobtanq)  60480  
49  teritruncated 2_{31} (totalq)  24192  
50  terirhombated 2_{31} (trilq)  120960  
51  demicelliprismated 3_{21} (hicpanq)  120960  
52  small teridemified 2_{31} (sethalq)  24192  
53  small cellated 3_{21} (scanq)  60480  
54  demiprismated 3_{21} (hipnaq)  80640  
55  terirhombated 3_{21} (tranq)  60480  
56  demicellirhombated 3_{21} (hocranq)  120960  
57  prismatorhombated 3_{21} (pranq)  120960  
58  small demirhombated 3_{21} (sharnaq)  60480  
59  teritruncated 3_{21} (tetanq)  15120  
60  demicellitruncated 3_{21} (hictanq)  60480  
61  prismatotruncated 3_{21} (potanq)  120960  
62  demitruncated 3_{21} (hotnaq)  24192  
63  great rhombated 3_{21} (granq)  24192  
64  great demified 2_{31} (gahlaq)  120960  
65  great demiprismated 2_{31} (gahplaq)  241920  
66  prismatotruncated 2_{31} (potlaq)  241920  
67  prismatorhombated 2_{31} (prolaq)  241920  
68  great rhombated 1_{32} (girlin)  241920  
69  celligreatorhombated 2_{31} (cagrilq)  362880  
70  cellidemitruncated 2_{31} (chotalq)  241920  
71  prismatotruncated 1_{32} (patlin)  362880  
72  biprismatorhombated 3_{21} (bipirnaq)  362880  
73  tritruncated 1_{32} (tatlin)  241920  
74  cellidemiprismatorhombated 2_{31} (chopralq)  362880  
75  great demibiprismated 3_{21} (ghobipnaq)  362880  
76  celliprismated 2_{31} (caplaq)  241920  
77  biprismatotruncated 3_{21} (boptanq)  362880  
78  great trirhombated 2_{31} (gatralaq)  241920  
79  terigreatorhombated 2_{31} (togrilq)  241920  
80  teridemitruncated 2_{31} (thotalq)  120960  
81  teridemirhombated 2_{31} (thorlaq)  241920  
82  celliprismated 3_{21} (capnaq)  241920  
83  teridemiprismatotruncated 2_{31} (thoptalq)  241920  
84  teriprismatorhombated 3_{21} (tapronaq)  362880  
85  demicelliprismatorhombated 3_{21} (hacpranq)  362880  
86  teriprismated 2_{31} (toplaq)  241920  
87  cellirhombated 3_{21} (cranq)  362880  
88  demiprismatorhombated 3_{21} (hapranq)  241920  
89  tericellitruncated 2_{31} (tectalq)  120960  
90  teriprismatotruncated 3_{21} (toptanq)  362880  
91  demicelliprismatotruncated 3_{21} (hecpotanq)  362880  
92  teridemitruncated 3_{21} (thotanq)  120960  
93  cellitruncated 3_{21} (catnaq)  241920  
94  demiprismatotruncated 3_{21} (hiptanq)  241920  
95  terigreatorhombated 3_{21} (tagranq)  120960  
96  demicelligreatorhombated 3_{21} (hicgarnq)  241920  
97  great prismated 3_{21} (gopanq)  241920  
98  great demirhombated 3_{21} (gahranq)  120960  
99  great prismated 2_{31} (gopalq)  483840  
100  great cellidemified 2_{31} (gechalq)  725760  
101  great birhombated 1_{32} (gebrolin)  725760  
102  prismatorhombated 1_{32} (prolin)  725760  
103  celliprismatorhombated 2_{31} (caprolaq)  725760  
104  great biprismated 2_{31} (gobpalq)  725760  
105  tericelliprismated 3_{21} (ticpanq)  483840  
106  teridemigreatoprismated 2_{31} (thegpalq)  725760  
107  teriprismatotruncated 2_{31} (teptalq)  725760  
108  teriprismatorhombated 2_{31} (topralq)  725760  
109  cellipriemsatorhombated 3_{21} (copranq)  725760  
110  tericelligreatorhombated 2_{31} (tecgrolaq)  725760  
111  tericellitruncated 3_{21} (tectanq)  483840  
112  teridemiprismatotruncated 3_{21} (thoptanq)  725760  
113  celliprismatotruncated 3_{21} (coptanq)  725760  
114  teridemicelligreatorhombated 3_{21} (thocgranq)  483840  
115  terigreatoprismated 3_{21} (tagpanq)  725760  
116  great demicellated 3_{21} (gahcnaq)  725760  
117  tericelliprismated laq (tecpalq)  483840  
118  celligreatorhombated 3_{21} (cogranq)  725760  
119  great demified 3_{21} (gahnq)  483840  
120  great cellated 2_{31} (gocalq)  1451520  
121  terigreatoprismated 2_{31} (tegpalq)  1451520  
122  tericelliprismatotruncated 3_{21} (tecpotniq)  1451520  
123  tericellidemigreatoprismated 2_{31} (techogaplaq)  1451520  
124  tericelligreatorhombated 3_{21} (tacgarnq)  1451520  
125  tericelliprismatorhombated 2_{31} (tecprolaq)  1451520  
126  great cellated 3_{21} (gocanq)  1451520  
127  great terated 3_{21} (gotanq)  2903040 
Regular and uniform honeycombs
There are five fundamental affine Coxeter groups and sixteen prismatic groups that generate regular and uniform tessellations in 6space:
#  Coxeter group  Coxeter diagram  Forms  

1  [3^{[7]}]  17  
2  [4,3^{4},4]  71  
3  h[4,3^{4},4] [4,3^{3},3^{1,1}]  95 (32 new)  
4  q[4,3^{4},4] [3^{1,1},3^{2},3^{1,1}]  41 (6 new)  
5  [3^{2,2,2}]  39 
Regular and uniform tessellations include:
 , 17 forms
 Uniform 6simplex honeycomb: {3^{[7]}}
 Uniform Cyclotruncated 6simplex honeycomb: t_{0,1}{3^{[7]}}
 Uniform Omnitruncated 6simplex honeycomb: t_{0,1,2,3,4,5,6,7}{3^{[7]}}
 Uniform 6simplex honeycomb: {3^{[7]}}
 , [4,3^{4},4], 71 forms
 Regular 6cube honeycomb, represented by symbols {4,3^{4},4},
 Regular 6cube honeycomb, represented by symbols {4,3^{4},4},
 , [3^{1,1},3^{3},4], 95 forms, 64 shared with , 32 new
 Uniform 6demicube honeycomb, represented by symbols h{4,3^{4},4} = {3^{1,1},3^{3},4},
=
 Uniform 6demicube honeycomb, represented by symbols h{4,3^{4},4} = {3^{1,1},3^{3},4},
 , [3^{1,1},3^{2},3^{1,1}], 41 unique ringed permutations, most shared with and , and 6 are new. Coxeter calls the first one a quarter 6cubic honeycomb.
= = = = = =
 : [3^{2,2,2}], 39 forms
 Uniform 2_{22} honeycomb: represented by symbols {3,3,3^{2,2}},
 Uniform t_{4}(2_{22}) honeycomb: 4r{3,3,3^{2,2}},
 Uniform 0_{222} honeycomb: {3^{2,2,2}},
 Uniform t_{2}(0_{222}) honeycomb: 2r{3^{2,2,2}},
 Uniform 2_{22} honeycomb: represented by symbols {3,3,3^{2,2}},
#  Coxeter group  CoxeterDynkin diagram  

1  x  [3^{[6]},2,∞]  
2  x  [4,3,3^{1,1},2,∞]  
3  x  [4,3^{3},4,2,∞]  
4  x  [3^{1,1},3,3^{1,1},2,∞]  
5  xx  [3^{[5]},2,∞,2,∞,2,∞]  
6  xx  [4,3,3^{1,1},2,∞,2,∞]  
7  xx  [4,3,3,4,2,∞,2,∞]  
8  xx  [3^{1,1,1,1},2,∞,2,∞]  
9  xx  [3,4,3,3,2,∞,2,∞]  
10  xxx  [4,3,4,2,∞,2,∞,2,∞]  
11  xxx  [4,3^{1,1},2,∞,2,∞,2,∞]  
12  xxx  [3^{[4]},2,∞,2,∞,2,∞]  
13  xxxx  [4,4,2,∞,2,∞,2,∞,2,∞]  
14  xxxx  [6,3,2,∞,2,∞,2,∞,2,∞]  
15  xxxx  [3^{[3]},2,∞,2,∞,2,∞,2,∞]  
16  xxxxx  [∞,2,∞,2,∞,2,∞,2,∞] 
Regular and uniform hyperbolic honeycombs
There are no compact hyperbolic Coxeter groups of rank 7, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 3 paracompact hyperbolic Coxeter groups of rank 7, each generating uniform honeycombs in 6space as permutations of rings of the Coxeter diagrams.
= [3,3^{[6]}]: 
= [3^{1,1},3,3^{2,1}]: 
= [4,3,3,3^{2,1}]: 
Notes on the Wythoff construction for the uniform 7polytopes
The reflective 7dimensional uniform polytopes are constructed through a Wythoff construction process, and represented by a CoxeterDynkin diagram, where each node represents a mirror. An active mirror is represented by a ringed node. Each combination of active mirrors generates a unique uniform polytope. Uniform polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may be named in two equally valid ways.
Here are the primary operators available for constructing and naming the uniform 7polytopes.
The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.
Operation  Extended Schläfli symbol 
Coxeter Dynkin diagram 
Description 

Parent  t_{0}{p,q,r,s,t,u}  Any regular 7polytope  
Rectified  t_{1}{p,q,r,s,t,u}  The edges are fully truncated into single points. The 7polytope now has the combined faces of the parent and dual.  
Birectified  t_{2}{p,q,r,s,t,u}  Birectification reduces cells to their duals.  
Truncated  t_{0,1}{p,q,r,s,t,u}  Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 7polytope. The 7polytope has its original faces doubled in sides, and contains the faces of the dual.  
Bitruncated  t_{1,2}{p,q,r,s,t,u}  Bitrunction transforms cells to their dual truncation.  
Tritruncated  t_{2,3}{p,q,r,s,t,u}  Tritruncation transforms 4faces to their dual truncation.  
Cantellated  t_{0,2}{p,q,r,s,t,u}  In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.  
Bicantellated  t_{1,3}{p,q,r,s,t,u}  In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.  
Runcinated  t_{0,3}{p,q,r,s,t,u}  Runcination reduces cells and creates new cells at the vertices and edges.  
Biruncinated  t_{1,4}{p,q,r,s,t,u}  Runcination reduces cells and creates new cells at the vertices and edges.  
Stericated  t_{0,4}{p,q,r,s,t,u}  Sterication reduces 4faces and creates new 4faces at the vertices, edges, and faces in the gaps.  
Pentellated  t_{0,5}{p,q,r,s,t,u}  Pentellation reduces 5faces and creates new 5faces at the vertices, edges, faces, and cells in the gaps.  
Hexicated  t_{0,6}{p,q,r,s,t,u}  Hexication reduces 6faces and creates new 6faces at the vertices, edges, faces, cells, and 4faces in the gaps. (expansion operation for 7polytopes)  
Omnitruncated  t_{0,1,2,3,4,5,6}{p,q,r,s,t,u}  All six operators, truncation, cantellation, runcination, sterication, pentellation, and hexication are applied. 
References
 Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
 T. Gosset: On the Regular and SemiRegular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
 A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
 H.S.M. Coxeter:
 H.S.M. Coxeter, M.S. LonguetHiggins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
 H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 http://www.wiley.com/WileyCDA/WileyTitle/productCd0471010030.html
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
 Klitzing, Richard. "7D uniform polytopes (polyexa)".
External links
 Polytope names
 Polytopes of Various Dimensions
 Multidimensional Glossary
 Glossary for hyperspace, George Olshevsky.