# Changes

## Element structure of dihedral group:D8

, 22:50, 21 June 2013
Interpretation as unitriangular matrix group
We view the dihedral group of order eight as a [[unitriangular matrix group of degree three]] over [[field:F2]], which is the group under multiplication of matrices of the form:
$\begin{pmatrix} 1 & a a_{12} & b a_{13} \\ 0 & 1 & c a_{23} \\ 0 & 0 & 1 \\\end{pmatrix}$
with the entries over the field of two elements. We compare with the general theory of the conjugacy class structure of the group $UT(3,q)$, where <matH>q</matH> is the field size. We denote by $p$ the prime number that is the characteristic of the field, so <matH>q[/itex] is a power of $p$.
Note that the letter $a$ used for matrix entries has no direct relation to the letter $a</matH> used for group elements of [itex]D_8$.
We compare with
{| class="sortable" border="1"
! Nature of conjugacy class !! Type of matrix !! Minimal polynomial !! Size of conjugacy class (generic $q$ ) !! Size of conjugacy class ($q = 2$) !! Number of such conjugacy classes (generic <matH>q[/itex]) !! Number of such conjugacy classes ($q = 2$) !! Total number of elements (generic $q$) !! Total number of elements ($q = 2$) !! List of conjugacy classes !! Order of elements in each such conjugacy class (generic $q$) !! Order of elements in each conjugacy class ($q = 2$) !! Type of matrix
|-
| identity element || $a = b = c = 0$ || $x t - 1$ || 1 || 1 || 1 || 1 || 1 || 1 || $\{ e \}$ || 1 || 1|| $a_{12} = a_{13} = a_{23}= 0$
|-
| non-identity element, but central (has Jordan blocks of size one and two respectively) || $a = c = 0$ $b \ne 0$ || $(x t - 1)^2$ || 1 || 1 || $q - 1$ || 1 || $q - 1$ || 1 || $\{ a^2 \}$ || $p$ || 2|| $a_{12} = a_{23} = 0$ $a_{13} \ne 0$
|-
| non-central, has Jordan blocks of size one and two respectively || $ac = 0$, but not both $a$ and $c$ are zero || $(x t - 1)^2$ || $q$ || 2 || $2q 2(q - 21)$ || 2 || $2q(q(2q - 21)$ || 4 || $\{ x, a^2x, \}, \{ ax, a^3x \}$ || $p$ || 2|| $a_{12}a_{23} = 0$, but not both $a_{12}$ and $a_{23}$ are zero
|-
| non-central, has Jordan block of size three || both $a$ and $c$ are nonzero || $(x t - 1)^3$ || $q$ || 2 || $(q - 1)^2$ || 1 ||$q(q - 1)^2$ || 2 || $\{ a, a^3 \}$ || $p$ if $p$ odd<br>4 if $p = 2$ || 4|| both $a_{12}$ and $a_{23}$ are nonzero
|-
! Total (--) !! -- !! -- !! -- !! -- !! $q^2 + q - 1$ !! 5 !! $q^3$ !! 8!! -- !! -- !! -- !! --
|}