# Difference between revisions of "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:

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

## 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. (2)
9. The subgroups $\langle a^2,x,y \rangle$, $\langle a^2, ax, y \rangle$, $\langle a^2, x, ay \rangle$, and $\langle a^2, ax, ay \rangle$. These are all normal and are related by outer automorphisms. Isomorphic to elementary abelian group of order eight. (4)
10. The subgroups $\langle a,x \rangle$ and $\langle a, xy \rangle$. These are both normal and are related by an outer automorphism. Isomorphic to dihedral group:D8. (2)
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)