# Changes

## Element structure of dihedral group:D8

, 02:21, 22 June 2011
Conjugacy class structure
{{conjugacy class structure facts to check against}}
<section begin="conjugacy and automorphism class structure"/>
{| class="sortable" border="1"
! Conjugacy class in terms of $a,x$ !! Geometric description of conjugacy class !! Conjugacy class as permutations !! Size of conjugacy class !! Order of elements in conjugacy class !! Centralizer of first element of class
{| class="sortable" border="1"
! Equivalence class under automorphisms in terms of $a,x$ !! Geometric description of equivalence class !! Equivalence class as permutations !! Size of equivalence class !! Number of conjugacy classes in it !! Size of each conjugacy class
|-
| $\! \{ e \}$ || identity element, does nothing || $\{ () \}$ || 1 || 1 || 1
|-
| $\! \{ a^2 \}$ || half turn || $\{ (1,3)(2,4) \}$ || 1 || 1 || 1
|-
| $\! \{ x, ax, a^2x, a^3x \}$ || reflections || $\{ (1,3), (2,4), (1,4)(2,3), (1,2)(3,4) \}$ || 4 || 2 || 2
|-
| $\! \{ a, a^3 \}$ || rotations by odd multiples of $\pi/2$ || $\{ (1,2,3,4), (1,4,3,2) \}$ || 2 || 1 || 2
|}
<section end="conjugacy and automorphism class structure"/>
===Convolution algebra on conjugacy classes===