octic group

The octic group also known as the $4^{th}$ dihedral group, is a non-Abelian group with eight elements. It is traditionally denoted by $D_4$. This group is defined by the presentation

or, equivalently, defined by the multiplication table

$\cdot$	$e$	$s$	$s^2$	$s^3$	$t$	$ts$	$t{s}^2$	$t{s}^3$
$e$	$e$	$s$	$s^2$	$s^3$	$t$	$ts$	$t{s}^2$	$t{s}^3$
$s$	$s$	$s^2$	$s^3$	$e$	$t{s}^3$	$t$	$ts$	$t{s}^2$
$s^2$	$s^2$	$s^3$	$e$	$s$	$t{s}^2$	$t{s}^3$	$t$	$ts$
$s^3$	$s^3$	$e$	$s$	$s^2$	$ts$	$t{s}^2$	$t{s}^3$	$t$
$t$	$t$	$ts$	$t{s}^2$	$t{s}^3$	$e$	$s$	$s^2$	$s^3$
$ts$	$ts$	$t{s}^2$	$t{s}^3$	$t$	$s^3$	$e$	$s$	$s^2$
$t{s}^2$	$t{s}^2$	$t{s}^3$	$t$	$ts$	$s^2$	$s^3$	$e$	$s$
$t{s}^3$	$t{s}^3$	$t$	$ts$	$t{s}^2$	$s$	$s^2$	$s^3$	$e$

where we have put each product $xy$ into row $x$ and column $y$. The lattice of the subgroups is given below:

where denotes the subgroup generated by $a_1,\mathrm{\dots },a_n$ and $\{b_1,\mathrm{\dots },b_n\}$ denotes the subgroup. Of those subgroups, the following are its proper normal subgroup: $\{e,s^2,t,t{s}^2\}$, , $\{e,s^2,st,ts\}$, and . In addition the center and commutator subgroup of the octic group is . It can also be shown that the automorphism of the octic group ($\mathrm{Aut}\mathit{}\mathrm{(}{D}_{\mathrm{4}}\mathrm{)}$) is isomorphic to itself($D_{\mathrm{4}}$).[PJ] An additional property is that the subgroup of the general linear group of dimension 2 over the real numbers generated by:[ 0 1 -1 0 ],[ 0 1 1 0 ]is isomorphic to the octic group.