Direct product of D8 and Z2

Definition
This group is defined in the following ways:


 * It is the external direct product of the dihedral group of order eight and the cyclic group of order two.
 * It is the member of family::generalized dihedral group corresponding to the direct product of Z4 and Z2.
 * It is the $$2$$-Sylow subgroup of the symmetric group of degree six.

A presentation for it is:

$$G := \langle a,x,y \mid a^4 = x^2 = y^2 = e, xax = a^{-1}, ya = ay, xy = yx \rangle$$.

Upto conjugacy
There are ten conjugacy classes:


 * 1) The identity element. (1)
 * 2) The element $$a^2$$. (1)
 * 3) The element $$y$$. (1)
 * 4) The element $$a^2y $$. (1)
 * 5) The two-element conjugacy class comprising $$a$$ and $$a^3$$. (2)
 * 6) The two-element conjugacy class comprising $$ay$$ and $$a^3y$$. (2)
 * 7) The two-element conjugacy class comprising $$x$$ and $$a^2x$$. (2)
 * 8) The two-element conjugacy class comprising $$xy$$ and $$a^2xy$$. (2)
 * 9) The two-element conjugacy class comprising $$ax$$ and $$a^3x$$. (2)
 * 10) The two-element conjugacy class comprising $$axy$$ and $$a^3xy$$. (2)

Upto automorphism
Under the action of the automorphism group, the conjugacy classes (3) and (4) are in the same orbit -- in other words, $$y$$ and $$a^2y$$ are related by an automorphism. The conjugacy classes (5) and (6) are equivalent, and the conjugacy classes (7)-(10) are equivalent. Thus, the equivalence classes have sizes $$1,1,2,4,8$$.

Subgroups
The group has the following subgroups:


 * 1) The trivial group. (1)
 * 2) The cyclic group $$\langle a^2 \rangle$$ of order two. This is central, and is also the set of squares. Isomorphic to subgroup::cyclic group:Z2. (1)
 * 3) The subgroups $$\langle y \rangle$$ and $$\langle a^2y \rangle$$. These are both central subgroups of order two, but are related by an outer automorphism. Isomorphic to subgroup::cyclic group:Z2. (2)
 * 4) The subgroups $$\langle x \rangle$$, $$\langle ax \rangle$$, $$\langle a^2x \rangle$$, $$\langle a^3x \rangle$$, $$\langle xy \rangle$$, $$\langle axy \rangle$$, $$\langle a^2xy \rangle$$, and $$\langle a^3xy \rangle$$. These are all related by automorphisms, and are all 2-subnormal subgroups. They come in four conjugacy classes, namely the class $$\langle x \rangle, \langle a^2x \rangle$$, the class $$\langle ax \rangle, \langle a^3x \rangle$$, the class $$\langle xy \rangle, \langle a^2xy$$, and the class $$\langle axy \rangle, \langle a^3xy \rangle$$. Isomorphic to subgroup::cyclic group:Z2. (8)
 * 5) The subgroup $$\langle a^2, y \rangle$$. This is the center, hence is a characteristic subgroup. Isomorphic to subgroup::Klein four-group. (1)
 * 6) The subgroups $$\langle a^2, x \rangle$$, $$\langle a^2, ax \rangle$$, $$\langle a^2, xy \rangle$$, and $$\langle a^2, axy \rangle$$. These are all normal subgroups, but are related by outer automorphisms. Isomorphic to subgroup::Klein four-group. (4)
 * 7) The subgroups $$\langle y, x \rangle$$, $$\langle y, ax \rangle$$, $$\langle y, a^2x \rangle$$, $$\langle y, a^3x$$, $$\langle a^2y, x \rangle$$, $$\langle a^2y, ax$$, $$\langle a^2y, a^2x \rangle$$, $$\langle a^2y, a^3x \rangle$$. These subgroups are all 2-subnormal subgroups and are related by outer automorphisms, and they come in four conjugacy classes of size two. Isomorphic to subgroup::Klein four-group. (8)
 * 8) The subgroups $$\langle a \rangle$$ and $$\langle ay \rangle$$. They are all normal and are related via outer automorphisms. Isomorphic to subgroup::cyclic group:Z4. (2)
 * 9) The subgroups $$\langle a^2,x,y \rangle$$ and $$\langle a^2, ax, y \rangle$$. These are normal and are related by outer automorphisms. Isomorphic to elementary abelian group of order eight. (2)
 * 10) The subgroups $$\langle a,x \rangle$$, $$\langle a, xy \rangle$$, $$\langle ay, x \rangle$$ and $$\langle ay, xy \rangle$$. These are both normal and are related by an outer automorphism. Isomorphic to subgroup::dihedral group:D8. (4)
 * 11) The subgroup $$\langle a, y \rangle$$. This is a characteristic subgroup. Isomorphic to subgroup::direct product of Z4 and Z2. (1)
 * 12) The whole group. (1)