Element structure of semidihedral group:SD16

This article gives specific information, namely, element structure, about a particular group, namely: semidihedral group:SD16.
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This article describes the structure of elements of the semidihedral group:SD16, which is given by the following presentation:

$\langle a,x\mid a^{8}=x^{2}=e,xax=a^{3}\rangle$

Here, $e$ denotes the identity element. Every element is of the form $a^{k},0\leq k\leq 7$ or $a^{k}x,0\leq k\leq 7$.

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
$\!\{a^{4}\}$	1	2	whole group
$\!\{a^{2},a^{6}\}$	2	4	$\langle a\rangle$ -- a cyclic subgroup of order $8$
$\!\{a,a^{3}\}$	2	8	$\langle a\rangle$ -- a cyclic subgroup of order $8$
$\!\{a^{5},a^{7}\}$	2	8	$\langle a\rangle$ -- a cyclic subgroup of order $8$
$\!\{x,a^{2}x,a^{4}x,a^{6}x\}$	4	2	$\!\{e,a^{4},x,a^{4}x\}$
$\!\{ax,a^{3}x,a^{5}x,a^{7}x\}$	4	4	$\!\{e,a^{4},ax,a^{5}x\}$

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
$\!\{a^{4}\}$	1	1	1
$\!\{a^{2},a^{6}\}$	2	1	2
$\!\{a,a^{3},a^{5},a^{7}\}$	4	2	2
$\!\{x,a^{2}x,a^{4}x,a^{6}x\}$	4	1	4
$\!\{ax,a^{3}x,a^{5}x,a^{7}x\}$	4	1	4

Order and power information

The graph below is a collapsed and trimmed version of the directed power graph of the group. Here, we collapse together all elements that generate the same cyclic subgroup, and an edge from one vertex to another is drawn if the latter is the square ofthe former. We omit the loop at the identity element.

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	$\!\{a^{4},x,a^{2}x,a^{4}x,a^{6}x\}$	5	2	6	3
4	$\!\{a^{2},a^{6},ax,a^{3}x,a^{5}x,a^{7}x\}$	6	2	12	5
8	$\!\{a,a^{3},a^{5},a^{7}\}$	4	2	16	7

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	$\!\{a,a^{3},a^{5},a^{7},x,ax,a^{2}x,a^{3}x,a^{4}x,a^{5}x,a^{6}x,a^{7}x\}$	12	4	16	7
2	$\!\{a^{2},a^{6}\}$	2	1	4	3
4	$\!\{a^{4}\}$	1	1	2	2
8	--	0	0	1	1
16	$\!\{e\}$	1	1	1	1