Dicyclic group
In group theory, a dicyclic group (notation Dic_{n} or Q_{4n}[1]) is a member of a class of nonabelian groups of order 4n (n > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2n, giving the name dicyclic. In the notation of exact sequences of groups, this extension can be expressed as:
Algebraic structure → Group theory Group theory 



Infinite dimensional Lie group

More generally, given any finite abelian group with an order2 element, one can define a dicyclic group.
Definition
For each integer n > 1, the dicyclic group Dic_{n} can be defined as the subgroup of the unit quaternions generated by
More abstractly, one can define the dicyclic group Dic_{n} as the group with the following presentation[2]
Some things to note which follow from this definition:
 x^{4} = 1
 x^{2}a^{k} = a^{k+n} = a^{k}x^{2}
 if j = ±1, then x^{j}a^{k} = a^{−k}x^{j}.
 a^{k}x^{−1} = a^{k−n}a^{n}x^{−1} = a^{k−n}x^{2}x^{−1} = a^{k−n}x.
Thus, every element of Dic_{n} can be uniquely written as a^{k}x^{j}, where 0 ≤ k < 2n and j = 0 or 1. The multiplication rules are given by
It follows that Dic_{n} has order 4n.[2]
When n = 2, the dicyclic group is isomorphic to the quaternion group Q. More generally, when n is a power of 2, the dicyclic group is isomorphic to the generalized quaternion group.[2]
Properties
For each n > 1, the dicyclic group Dic_{n} is a nonabelian group of order 4n. (For the degenerate case n = 1, the group Dic_{1} is the cyclic group C_{4}, which is not considered dicyclic.)
Let A = ⟨a⟩ be the subgroup of Dic_{n} generated by a. Then A is a cyclic group of order 2n, so [Dic_{n}:A] = 2. As a subgroup of index 2 it is automatically a normal subgroup. The quotient group Dic_{n}/A is a cyclic group of order 2.
Dic_{n} is solvable; note that A is normal, and being abelian, is itself solvable.
Binary dihedral group
The dicyclic group is a binary polyhedral group — it is one of the classes of subgroups of the Pin group Pin_{−}(2), which is a subgroup of the Spin group Spin(3) — and in this context is known as the binary dihedral group.
The connection with the binary cyclic group C_{2n}, the cyclic group C_{n}, and the dihedral group Dih_{n} of order 2n is illustrated in the diagram at right, and parallels the corresponding diagram for the Pin group.
There is a superficial resemblance between the dicyclic groups and dihedral groups; both are a sort of "mirroring" of an underlying cyclic group. But the presentation of a dihedral group would have x^{2} = 1, instead of x^{2} = a^{n}; and this yields a different structure. In particular, Dic_{n} is not a semidirect product of A and ⟨x⟩, since A ∩ ⟨x⟩ is not trivial.
The dicyclic group has a unique involution (i.e. an element of order 2), namely x^{2} = a^{n}. Note that this element lies in the center of Dic_{n}. Indeed, the center consists solely of the identity element and x^{2}. If we add the relation x^{2} = 1 to the presentation of Dic_{n} one obtains a presentation of the dihedral group Dih_{2n}, so the quotient group Dic_{n}/<x^{2}> is isomorphic to Dih_{n}.
There is a natural 2to1 homomorphism from the group of unit quaternions to the 3dimensional rotation group described at quaternions and spatial rotations. Since the dicyclic group can be embedded inside the unit quaternions one can ask what the image of it is under this homomorphism. The answer is just the dihedral symmetry group Dih_{n}. For this reason the dicyclic group is also known as the binary dihedral group. Note that the dicyclic group does not contain any subgroup isomorphic to Dih_{n}.
The analogous preimage construction, using Pin_{+}(2) instead of Pin_{−}(2), yields another dihedral group, Dih_{2n}, rather than a dicyclic group.
Generalizations
Let A be an abelian group, having a specific element y in A with order 2. A group G is called a generalized dicyclic group, written as Dic(A, y), if it is generated by A and an additional element x, and in addition we have that [G:A] = 2, x^{2} = y, and for all a in A, x^{−1}ax = a^{−1}.
Since for a cyclic group of even order, there is always a unique element of order 2, we can see that dicyclic groups are just a specific type of generalized dicyclic group.
See also
References
 Nicholson, W. Keith (1999). Introduction to Abstract Algebra (2nd ed.). New York: John Wiley & Sons, Inc. p. 449. ISBN 0471331090.
 Roman, Steven (2011). Fundamentals of Group Theory: An Advanced Approach. Springer. pp. 347–348. ISBN 9780817683016.
 Coxeter, H. S. M. (1974), "7.1 The Cyclic and Dicyclic groups", Regular Complex Polytopes, Cambridge University Press, pp. 74–75.
 Coxeter, H. S. M.; Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: SpringerVerlag. ISBN 0387092129.