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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{2}=b^{4}=c^{4}=d^{12}=[a,c]=[c,d]=1, b^{a}=b^{3}, d^{a}=c^{2}d^{7}, c^{b}=c^{3}d^{9}, d^{b}=c^{2}d^{5} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$D_4^2:S_3$ 1A $1$ $1$ $D_4^2:S_3$ 1A 1A $1$
$D_4^2:S_3$ 2A $2$ $1$ $D_4^2:S_3$ 1A 2A $d^{6}$
$D_4^2:S_3$ 2B $2$ $2$ $C_3\times D_4^2$ 1A 2B $c^{2}$
$D_4^2:S_3$ 2C $2$ $4$ $C_{12}:C_2^3$ 1A 2C $a$
$D_4^2:S_3$ 2D $2$ $4$ $C_{12}:C_2^3$ 1A 2D $ad^{6}$
$D_4^2:S_3$ 2E $2$ $4$ $C_{12}:C_2^3$ 1A 2E $acd^{9}$
$D_4^2:S_3$ 2F $2$ $4$ $C_{12}:C_2^3$ 1A 2F $ac^{3}d^{9}$
$D_4^2:S_3$ 2G $2$ $4$ $C_2^4:S_3$ 1A 2G $b^{2}$
$D_4^2:S_3$ 2H $2$ $4$ $C_2^4:S_3$ 1A 2H $b^{2}d^{9}$
$D_4^2:S_3$ 2I $2$ $8$ $C_2^3\times C_6$ 1A 2I $b^{2}c$
$D_4^2:S_3$ 2J $2$ $24$ $C_2\times D_4$ 1A 2J $abc^{2}d^{11}$
$D_4^2:S_3$ 3A $3$ $2$ $C_3\times D_4^2$ 3A 1A $d^{8}$
$D_4^2:S_3$ 4A $4$ $4$ $D_4\times C_{12}$ 2B 4A $c^{3}$
$D_4^2:S_3$ 4B $4$ $4$ $D_4\times C_{12}$ 2B 4B $c^{3}d^{6}$
$D_4^2:S_3$ 4C $4$ $4$ $D_{12}:C_4$ 2A 4C $d^{3}$
$D_4^2:S_3$ 4D $4$ $8$ $C_2^2\times C_{12}$ 2B 4D $ac^{2}d^{9}$
$D_4^2:S_3$ 4E $4$ $8$ $C_2^2\times C_{12}$ 2B 4E $ac^{3}$
$D_4^2:S_3$ 4F $4$ $24$ $D_4:C_2$ 2A 4F $abc^{2}$
$D_4^2:S_3$ 4G $4$ $48$ $C_2\times C_4$ 2G 4G $bc^{2}d^{4}$
$D_4^2:S_3$ 4H $4$ $48$ $C_2\times C_4$ 2H 4H $bc^{3}d^{5}$
$D_4^2:S_3$ 6A $6$ $2$ $C_3\times D_4^2$ 3A 2A $d^{2}$
$D_4^2:S_3$ 6B1 $6$ $2$ $C_3\times D_4^2$ 3A 2B $c^{2}d^{8}$
$D_4^2:S_3$ 6B-1 $6$ $2$ $C_3\times D_4^2$ 3A 2B $c^{2}d^{4}$
$D_4^2:S_3$ 6C1 $6$ $4$ $C_{12}:C_2^3$ 3A 2C $ad^{4}$
$D_4^2:S_3$ 6C-1 $6$ $4$ $C_{12}:C_2^3$ 3A 2C $ad^{8}$
$D_4^2:S_3$ 6D1 $6$ $4$ $C_{12}:C_2^3$ 3A 2D $ad^{2}$
$D_4^2:S_3$ 6D-1 $6$ $4$ $C_{12}:C_2^3$ 3A 2D $ac^{2}d^{4}$
$D_4^2:S_3$ 6E1 $6$ $4$ $C_{12}:C_2^3$ 3A 2E $acd$
$D_4^2:S_3$ 6E-1 $6$ $4$ $C_{12}:C_2^3$ 3A 2E $acd^{5}$
$D_4^2:S_3$ 6F1 $6$ $4$ $C_{12}:C_2^3$ 3A 2F $ac^{3}d$
$D_4^2:S_3$ 6F-1 $6$ $4$ $C_{12}:C_2^3$ 3A 2F $ab^{2}cd^{4}$
$D_4^2:S_3$ 6G $6$ $8$ $C_2^3\times C_6$ 3A 2G $b^{2}d^{4}$
$D_4^2:S_3$ 6H $6$ $8$ $C_2^3\times C_6$ 3A 2H $b^{2}d$
$D_4^2:S_3$ 6I1 $6$ $8$ $C_2^3\times C_6$ 3A 2I $b^{2}cd^{4}$
$D_4^2:S_3$ 6I-1 $6$ $8$ $C_2^3\times C_6$ 3A 2I $b^{2}cd^{2}$
$D_4^2:S_3$ 8A $8$ $48$ $C_8$ 4C 8A $abcd^{9}$
$D_4^2:S_3$ 12A1 $12$ $4$ $D_4\times C_{12}$ 6B1 4A $cd^{4}$
$D_4^2:S_3$ 12A-1 $12$ $4$ $D_4\times C_{12}$ 6B-1 4A $cd^{8}$
$D_4^2:S_3$ 12B1 $12$ $4$ $D_4\times C_{12}$ 6B-1 4B $cd^{2}$
$D_4^2:S_3$ 12B-1 $12$ $4$ $D_4\times C_{12}$ 6B1 4B $cd^{5}$
$D_4^2:S_3$ 12C $12$ $8$ $C_4\times C_{12}$ 6A 4C $d$
$D_4^2:S_3$ 12D1 $12$ $8$ $C_2^2\times C_{12}$ 6B1 4D $ad$
$D_4^2:S_3$ 12D-1 $12$ $8$ $C_2^2\times C_{12}$ 6B-1 4D $ad^{5}$
$D_4^2:S_3$ 12E1 $12$ $8$ $C_2^2\times C_{12}$ 6B1 4E $acd^{4}$
$D_4^2:S_3$ 12E-1 $12$ $8$ $C_2^2\times C_{12}$ 6B-1 4E $acd^{2}$
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