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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{2}=b^{2}=c^{4}=d^{12}=[a,b]=[a,c]=[c,d]=1, d^{a}=c^{2}d^{11}, c^{b}=c^{3}, d^{b}=d^{7} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_3:D_4^2$ 1A $1$ $1$ $C_3:D_4^2$ 1A 1A $1$
$C_3:D_4^2$ 2A $2$ $1$ $C_3:D_4^2$ 1A 2A $c^{2}$
$C_3:D_4^2$ 2B $2$ $1$ $C_3:D_4^2$ 1A 2B $c^{2}d^{6}$
$C_3:D_4^2$ 2C $2$ $1$ $C_3:D_4^2$ 1A 2C $d^{6}$
$C_3:D_4^2$ 2D $2$ $4$ $C_2^2\times D_6$ 1A 2D $b$
$C_3:D_4^2$ 2E $2$ $4$ $C_2^2\times D_6$ 1A 2E $bd^{9}$
$C_3:D_4^2$ 2F $2$ $4$ $C_2^2\times D_6$ 1A 2F $bc$
$C_3:D_4^2$ 2G $2$ $4$ $C_2^2\times D_6$ 1A 2G $bcd^{9}$
$C_3:D_4^2$ 2H $2$ $6$ $C_2^2\times D_4$ 1A 2H $a$
$C_3:D_4^2$ 2I $2$ $6$ $C_2^2\times D_4$ 1A 2I $ad^{2}$
$C_3:D_4^2$ 2J $2$ $6$ $C_2^2\times D_4$ 1A 2J $ab$
$C_3:D_4^2$ 2K $2$ $6$ $C_2^2\times D_4$ 1A 2K $acd$
$C_3:D_4^2$ 2L $2$ $6$ $C_2^2\times D_4$ 1A 2L $abd^{2}$
$C_3:D_4^2$ 2M $2$ $6$ $C_2^2\times D_4$ 1A 2M $abc$
$C_3:D_4^2$ 2N $2$ $6$ $C_2^2\times D_4$ 1A 2N $acd^{3}$
$C_3:D_4^2$ 2O $2$ $6$ $C_2^2\times D_4$ 1A 2O $abcd^{2}$
$C_3:D_4^2$ 3A $3$ $2$ $C_{12}:D_4$ 3A 1A $d^{8}$
$C_3:D_4^2$ 4A $4$ $2$ $C_4\times D_{12}$ 2A 4A $c^{3}$
$C_3:D_4^2$ 4B $4$ $2$ $C_4\times D_{12}$ 2A 4B $c^{3}d^{6}$
$C_3:D_4^2$ 4C $4$ $2$ $C_4\times D_{12}$ 2B 4C $c^{3}d^{3}$
$C_3:D_4^2$ 4D $4$ $2$ $C_4\times D_{12}$ 2B 4D $cd^{3}$
$C_3:D_4^2$ 4E $4$ $4$ $C_4\times C_{12}$ 2C 4E $d^{3}$
$C_3:D_4^2$ 4F $4$ $12$ $C_2^2\times C_4$ 2A 4F $ad$
$C_3:D_4^2$ 4G $4$ $12$ $C_2^2\times C_4$ 2A 4G $ac$
$C_3:D_4^2$ 4H $4$ $12$ $C_2^2\times C_4$ 2B 4H $abd$
$C_3:D_4^2$ 4I $4$ $12$ $C_2^2\times C_4$ 2B 4I $abcd$
$C_3:D_4^2$ 6A $6$ $2$ $C_{12}:D_4$ 3A 2A $c^{2}d^{8}$
$C_3:D_4^2$ 6B $6$ $2$ $C_{12}:D_4$ 3A 2B $c^{2}d^{2}$
$C_3:D_4^2$ 6C $6$ $2$ $C_{12}:D_4$ 3A 2C $d^{2}$
$C_3:D_4^2$ 6D $6$ $8$ $C_2^2\times C_6$ 3A 2D $bd^{4}$
$C_3:D_4^2$ 6E $6$ $8$ $C_2^2\times C_6$ 3A 2E $bd$
$C_3:D_4^2$ 6F $6$ $8$ $C_2^2\times C_6$ 3A 2F $bcd^{4}$
$C_3:D_4^2$ 6G $6$ $8$ $C_2^2\times C_6$ 3A 2G $bcd$
$C_3:D_4^2$ 12A $12$ $4$ $C_4\times C_{12}$ 6A 4A $cd^{4}$
$C_3:D_4^2$ 12B $12$ $4$ $C_4\times C_{12}$ 6A 4B $cd^{2}$
$C_3:D_4^2$ 12C $12$ $4$ $C_4\times C_{12}$ 6B 4C $cd$
$C_3:D_4^2$ 12D $12$ $4$ $C_4\times C_{12}$ 6B 4D $c^{3}d$
$C_3:D_4^2$ 12E1 $12$ $4$ $C_4\times C_{12}$ 6C 4E $d$
$C_3:D_4^2$ 12E-1 $12$ $4$ $C_4\times C_{12}$ 6C 4E $d^{5}$
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