# Changes

## Element structure of dihedral group:D8

, 16:45, 28 March 2012
Interpretation as unitriangular matrix group
| non-identity element, but central (has Jordan blocks of size one and two respectively) || [itex](x - 1)^2[/itex] || 1 || 1 || [itex]q - 1[/itex] || 1 || [itex]q - 1[/itex] || 1 || [itex]\{ a^2 \}[/itex] || [itex]p[/itex] || 2 || [itex]a_{12} = a_{23} = 0[/itex] [itex]a_{13} \ne 0[/itex]
|-
| non-central, has Jordan blocks of size one and two respectively || [itex](x - 1)^2[/itex] || [itex]q[/itex] || 2 || [itex]2q 2(q - 21)[/itex] || 2 || [itex]2q(q - 1)[/itex] || 4 || [itex]\{ x, a^2x, \}, \{ ax, a^3x \}[/itex] || [itex]p[/itex] || 2 || [itex]a_{12}a_{23} = 0[/itex], but not both [itex]a_{12}[/itex] and [itex]a_{23}[/itex] are zero
|-
| non-central, has Jordan block of size three || [itex](x - 1)^3[/itex] || [itex]q[/itex] || 2 || [itex](q - 1)^2[/itex] || 1 ||[itex]q(q - 1)^2[/itex] || 2 || [itex]\{ a, a^3 \}[/itex] || [itex]p[/itex] if [itex]p[/itex] odd<br>4 if [itex]p = 2[/itex] || 4 || both [itex]a_{12}[/itex] and [itex]a_{23}[/itex] are nonzero