Generalized quaternion group:Q16

Definition
The group $$Q_{16}$$, sometimes termed the generalized quaternion group of order $$16$$, is a generalized quaternion group. It can be described by the following presentation:

$$G := \langle a,b,c \mid a^4 = b^2 = c^2 = abc \rangle$$.

Note that $$c = ab = ba^{-1}$$ from these relations, and $$bab^{-1} = a^{-1}$$. This in turn forces that $$b^2 = b(b^2)b^{-1} = ba^4b^{-1} = a^{-4} = b^{-2}$$, forcing $$b^2 = a^4 = c^2 = abc$$ to have order two. We shall denote this element of order two, which is clearly central, as $$z$$.

We can thus use an alternative presentation that requires only two generators:

$$G := \langle a,b \mid a^4 = b^2 = abab \rangle$$

Subgroups

 * 1) The trivial subgroup. Isomorphic to subgroup::trivial group. (1)
 * 2) The center, which is a subgroup of order two, generated by $$z = a^4 = b^2 = c^2$$. Isomorphic to subgroup::cyclic group:Z2. (1)
 * 3) The cyclic subgroup of order four generated by $$a^2$$. Isomorphic to subgroup::cyclic group:Z4. (1)
 * 4) The four cyclic subgroups of order four, namely: $$\langle b \rangle$$, $$\langle ab \rangle$$, $$\langle a^2b \rangle$$ and $$\langle a^3b\rangle$$. These come in two conjugacy classes of 2-subnormal subgroups, one conjugacy class comprising $$\langle ab \rangle$$ and $$\langle a^3b \rangle$$ and the other comprising $$\langle b \rangle$$ and $$\langle a^2b \rangle$$. Isomorphic to subgroup::cyclic group:Z4. (4)
 * 5) The cyclic subgroup of order eight, generated by $$a$$. This is characteristic; in fact, it equals the centralizer of commutator subgroup. Isomorphic to subgroup::cyclic group:Z8. (1)
 * 6) Two quaternion groups of order eight, namely $$\langle a^2,b \rangle$$ and $$\langle a^2, ab \rangle$$. Isomorphic to subgroup::quaternion group. (2)
 * 7) The whole group. (1)

Description by presentation
gap> F := FreeGroup(2);  gap> G := F/[F.1^8,F.2^2*F.1^(-4),F.2*F.1*F.2^(-1)*F.1];  gap> IdGroup(G); [ 16, 9 ]