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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{2}=c^{8}=d^{8}=[c,d]=1, b^{4}=d^{4}, b^{a}=b^{3}c^{7}d, c^{a}=cd^{3}, d^{a}=d^{7}, c^{b}=c^{7}d^{5}, d^{b}=c^{6}d \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P
$C_8^2.D_4$ 1A $1$ $1$ $C_8^2.D_4$ 1A $1$
$C_8^2.D_4$ 2A $2$ $1$ $C_8^2.D_4$ 1A $d^{4}$
$C_8^2.D_4$ 2B $2$ $2$ $C_8.Q_{32}$ 1A $c^{4}$
$C_8^2.D_4$ 2C $2$ $16$ $C_2\times Q_{16}$ 1A $a$
$C_8^2.D_4$ 4A $4$ $4$ $C_8:C_{16}$ 2B $c^{2}$
$C_8^2.D_4$ 4B $4$ $4$ $C_8\wr C_2$ 2A $d^{2}$
$C_8^2.D_4$ 4C $4$ $4$ $C_8:C_{16}$ 2B $c^{2}d^{4}$
$C_8^2.D_4$ 4D $4$ $16$ $D_8:C_2$ 2A $ac^{4}$
$C_8^2.D_4$ 4E $4$ $32$ $\OD_{16}$ 2A $b^{2}c^{6}d$
$C_8^2.D_4$ 4F $4$ $32$ $D_4:C_2$ 2A $b^{2}cd^{4}$
$C_8^2.D_4$ 8A1 $8$ $4$ $C_8:C_{16}$ 4A $cd^{4}$
$C_8^2.D_4$ 8A3 $8$ $4$ $C_8:C_{16}$ 4A $c^{5}d^{4}$
$C_8^2.D_4$ 8B1 $8$ $4$ $C_8:C_{16}$ 4A $c^{7}$
$C_8^2.D_4$ 8B3 $8$ $4$ $C_8:C_{16}$ 4A $c^{5}$
$C_8^2.D_4$ 8C1 $8$ $4$ $C_8\wr C_2$ 4B $c^{6}d$
$C_8^2.D_4$ 8C3 $8$ $4$ $C_8\wr C_2$ 4B $c^{2}d^{3}$
$C_8^2.D_4$ 8D $8$ $8$ $C_8^2$ 4B $c^{4}d$
$C_8^2.D_4$ 8E1 $8$ $8$ $C_8^2$ 4C $cd^{2}$
$C_8^2.D_4$ 8E3 $8$ $8$ $C_8^2$ 4C $cd^{6}$
$C_8^2.D_4$ 8F $8$ $32$ $C_2\times C_8$ 4B $ab^{2}c^{5}d$
$C_8^2.D_4$ 8G1 $8$ $64$ $C_8$ 4E $b^{3}c^{7}$
$C_8^2.D_4$ 8G-1 $8$ $64$ $C_8$ 4E $bc^{7}d$
$C_8^2.D_4$ 16A1 $16$ $16$ $C_2\times C_{16}$ 8B1 $ab$
$C_8^2.D_4$ 16A-1 $16$ $16$ $C_2\times C_{16}$ 8B1 $abc$
$C_8^2.D_4$ 16A3 $16$ $16$ $C_2\times C_{16}$ 8B3 $abcd$
$C_8^2.D_4$ 16A-3 $16$ $16$ $C_2\times C_{16}$ 8B3 $abd^{2}$
$C_8^2.D_4$ 16A5 $16$ $16$ $C_2\times C_{16}$ 8B3 $abd^{6}$
$C_8^2.D_4$ 16A-5 $16$ $16$ $C_2\times C_{16}$ 8B3 $abcd^{2}$
$C_8^2.D_4$ 16A7 $16$ $16$ $C_2\times C_{16}$ 8B1 $ab^{3}$
$C_8^2.D_4$ 16A-7 $16$ $16$ $C_2\times C_{16}$ 8B1 $abd^{4}$
$C_8^2.D_4$ 16B1 $16$ $32$ $C_{16}$ 8C1 $ac^{3}d^{3}$
$C_8^2.D_4$ 16B3 $16$ $32$ $C_{16}$ 8C3 $acd^{4}$
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