Element structure of nontrivial semidirect product of Z4 and Z4

This article gives specific information, namely, element structure, about a particular group, namely: nontrivial semidirect product of Z4 and Z4.
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This article gives information on the element structure of the group:

$G:=\langle x,y\mid x^{4}=y^{4}=e,yxy^{-1}=x^{3}\rangle$

where $e$ denotes the identity element.

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
$\!\{x^{2}\}$	1	2	whole group
$\!\{y^{2}\}$	1	2	whole group
$\!\{x^{2}y^{2}\}$	1	2	whole group
$\!\{x,x^{3}\}$	2	4	$\langle x,y^{2}\rangle$ -- isomorphic to direct product of Z4 and Z2
$\!\{y,x^{2}y\}$	2	4	$\langle x^{2},y\rangle$ -- isomorphic to direct product of Z4 and Z2
$\!\{y^{3},x^{2}y^{3}\}$	2	4	$\langle x^{2},y\rangle$ -- isomorphic to direct product of Z4 and Z2
$\!\{xy^{2},x^{3}y^{2}\}$	2	4	$\langle x,y^{2}\rangle$ -- isomorphic to direct product of Z4 and Z2
$\!\{xy,x^{3}y\}$	2	4	$\langle x^{2},xy\rangle$ -- isomorphic to direct product of Z4 and Z2
$\!\{xy^{3},x^{3}y^{3}\}$	2	4	$\langle x^{2},xy\rangle$ -- isomorphic to direct product of Z4 and Z2

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
$\!\{x^{2}\}$	1	1	1
$\!\{y^{2}\}$	1	1	1
$\!\{x^{2}y^{2}\}$	1	1	1
$\!\{x,x^{3},xy^{2},x^{3}y^{2}\}$	4	2	2
$\!\{y,x^{2}y,y^{3},x^{2}y^{3},xy,x^{3}y,xy^{3},x^{3}y^{3}\}$	8	4	2

Order and power information

Directed power graph

Below is a collapsed version of the directed power graph of the group. Each node represets an equivalence class of elements that all generate the same cyclic subgroup. There is an edge from one vertex to another if the latter is the square of the former. A dashed edge means that the latter is an odd power of the former. We remove all the loops.

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	$\!\{x^{2},y^{2},x^{2}y^{2}\}$	3	3	4	4
4	$\!\{x,x^{3},y,x^{2}y,y^{3},x^{2}y^{3},xy,x^{3}y,xy^{2},x^{3}y^{2},xy^{3},x^{3}y^{3}\}$	12	6	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	$\!\{x^{2}y^{2},x,x^{3},y,x^{2}y,y^{3},x^{2}y^{3},xy,x^{3}y,xy^{2},x^{3}y^{2},xy^{3},x^{3}y^{3}\}$	13	7	16	10
2	$\!\{x^{2},y^{2}\}$	2	2	3	3
4	--	0	0	1	1
8	--	0	0	1	1
16	$\!\{e\}$	1	1	1	1