Element structure of direct product of D8 and Z2

This article gives specific information, namely, element structure, about a particular group, namely: direct product of D8 and Z2.
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We use the following presentation, where $e$ denotes the identity element:

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

Conjugacy class structure

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

Conjugacy class	Size of conjugacy class	Order of elements in conjugacy class	Centralizer of first element of class
$\!\{e\}$	1	1	whole group
$\!\{a^{2}\}$	1	2	whole group
$\!\{y\}$	1	2	whole group
$\!\{a^{2}y\}$	1	2	whole group
$\!\{a,a^{3}\}$	2	4	$\langle a,y\rangle$ -- isomorphic to direct product of Z4 and Z2
$\!\{ay,a^{3}y\}$	2	4	$\langle a,y\rangle$ -- isomorphic to direct product of Z4 and Z2
$\!\{x,a^{2}x\}$	2	2	$\langle a^{2},x,y\rangle$ -- isomorphic to elementary abelian group:E8
$\!\{xy,a^{2}xy\}$	2	2	$\langle a^{2},x,y\rangle$ -- isomorphic to elementary abelian group:E8
$\!\{ax,a^{3}x\}$	2	2	$\langle a^{2},ax,y\rangle$ -- isomorphic to elementary abelian group:E8
$\!\{axy,a^{3}xy\}$	2	2	$\langle a^{2},ax,y\rangle$ -- isomorphic to elementary abelian group:E8

The equivalence classes up to automorphisms are:

Equivalence class under automorphisms	Size of equivalence class	Number of conjugacy classes in it	Size of each conjugacy class
$\!\{e\}$	1	1	1
$\!\{a^{2}\}$	1	1	1
$\!\{y,a^{2}y\}$	2	2	1
$\!\{a,a^{3},ay,a^{3}y\}$	4	2	2
$\!\{x,ax,a^{2}x,a^{3}x,xy,axy,a^{2}xy,a^{3}xy\}$	8	4	2

Order and power information

FACTS TO CHECK AGAINST:

ORDER STATISTICS (cf. order statistics, order statistics-equivalent finite groups): number of nth roots is a multiple of n | Finite abelian groups with the same order statistics are isomorphic | Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring | Frobenius conjecture on nth roots

1-ISOMORPHISM (cf. 1-isomorphic groups): Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring | order statistics-equivalent not implies 1-isomorphic

Order statistics

To compare and contrast the order statistics of this group with other groups of the same order, refer element structure of groups of order 16#Order statistics

Number	Elements of order exactly that number	Number of such elements	Number of conjugacy classes of such elements	Number of elements whose order divides that number	Number of conjugacy classes whose element order divides that number
1	$\!\{e\}$	1	1	1	1
2	$\!\{a^{2},y,a^{2}y,x,ax,a^{2}x,a^{3}x,xy,axy,a^{2}xy,a^{3}xy\}$	11	7	12	8
4	$\!\{a,a^{3},ay,a^{3}y\}$	4	2	16	10

Power statistics

Number $d$	$d^{th}$ powers that are not $k^{th}$ powers for any larger divisor $k$ of the group order	Number of such elements	Number of conjugacy classes of such elements	Number of $d^{th}$ powers	Number of conjugacy classes of $d^{th}$ powers
1	$\!\{y,a^{2}y,x,ax,a^{2}x,a^{3}x,xy,axy,a^{2}xy,a^{3}xy,a,a^{3},ay,a^{3}y\}$	14	8	16	10
2	$\!\{a^{2}\}$	1	1	2	2
4	--	0	0	1	1
8	--	0	0	1	1
16	$\!\{e\}$	1	1	1	1