# 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

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

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