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