Direct product of D8 and Z2

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This particular group is a finite group of order: 16

Definition

This group is defined as the external direct product of the dihedral group of order eight and the cyclic group of order two.

A presentation for it is:

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

Elements

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 $1,1,2,4,8$.

Group properties

Property Satisfied Explanation Comment
Abelian group No $a$ and $x$ don't commute
Nilpotent group Yes Prime power order implies nilpotent
Metacyclic group No
Supersolvable group Yes Finite nilpotent implies supersolvable
Solvable group Yes Nilpotent implies solvable
T-group No $\langle x \rangle \triangleleft \langle a^2,x \rangle$, which is normal, but $\langle x \rangle$ is not normal Smallest example for normality is not transitive.
Monolithic group No Center is Klein four-group, any subgroup of order two is minimal normal.
One-headed group No Distinct maximal subgroups of order eight.
SC-group No
ACIC-group Yes Every automorph-conjugate subgroup is characteristic
Rational group Yes Any two elements that generate the same cyclic group are conjugate Thus, all characters are integer-valued.
Rational-representation group Yes All representations over characteristic zero are realized over the rationals. Contrast with quaternion group, that is rational but not rational-representation.

Subgroups

Further information: Subgroup structure of direct product of D8 and Z2

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 cyclic group:Z2. (1)
3. The subgroups $\langle y \rangle$ and $\rangle a^2y$. These are both central subgroups of order two, but are related by an outer automorphism. Isomorphic to cyclic group:Z2. (2)
4. The subgroups $\langle x \rangle$, $\langle ax \rangle$, $\langle a^2x \rangle$, $\langle a^3x \rangle$, $\langle xy \rangle$, $\langle axy \rangle$, $\langle a^2xy$, and $\langle a^3xy$. 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 cyclic group:Z2. (8)
5. The subgroup $\langle a^2, y \rangle$. This is the center, hence is a characteristic subgroup. Isomorphic to 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 Klein four-group. (4)
7. The subgroups $\langle y, x \rangle$, $\langle y, ax \rangle$, $y, a^2x$, $\langle y, a^3x$, $\langle a^2y, x \rangle$, $\langle a^2y, ax$, $\langle a^2y, a^2x \rangle$, $\langle a^2y, a^3x \rangle$. 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 Klein four-group. (8)
8. The subgroups $\langle a \rangle$ and $\langle ay \rangle$. They are both normal and are related via an outer automorphism. Isomorphic to 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 dihedral group:D8. (4)
11. The subgroup $\langle a, y \rangle$. This is a characteristic subgroup. Isomorphic to direct product of Z4 and Z2. (1)
12. The whole group. (1)

GAP implementation

Group ID

The ID of this group in GAP's list of groups of order $16$ is $11$. The group can be described using GAP's SmallGroup function as:

SmallGroup(16,11)

Other descriptions

It can be described as a direct product:

DirectProduct(DihedralGroup(8),CyclicGroup(2))