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

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_6^2:D_6$ 1A $1$ $1$ $C_6^2:D_6$ 1A 1A $1$
$C_6^2:D_6$ 2A $2$ $3$ $D_6:D_6$ 1A 2A $d^{3}$
$C_6^2:D_6$ 2B $2$ $9$ $C_2\times S_4$ 1A 2B $b^{3}c^{2}$
$C_6^2:D_6$ 2C $2$ $18$ $C_2\times D_6$ 1A 2C $ab^{4}c^{2}$
$C_6^2:D_6$ 2D $2$ $18$ $C_2\times D_6$ 1A 2D $abc^{2}d^{4}$
$C_6^2:D_6$ 2E $2$ $27$ $C_2\times D_4$ 1A 2E $b^{3}cd^{5}$
$C_6^2:D_6$ 3A $3$ $2$ $C_3^2:S_4$ 3A 1A $c^{4}$
$C_6^2:D_6$ 3B $3$ $6$ $C_6\wr C_2$ 3B 1A $c^{4}d^{2}$
$C_6^2:D_6$ 3C $3$ $24$ $C_3\times S_3$ 3C 1A $b^{2}d^{3}$
$C_6^2:D_6$ 3D $3$ $48$ $C_3^2$ 3D 1A $b^{2}cd$
$C_6^2:D_6$ 4A $4$ $18$ $C_4\times S_3$ 2A 4A $ac^{5}d^{3}$
$C_6^2:D_6$ 4B $4$ $18$ $C_4\times S_3$ 2A 4B $abd^{3}$
$C_6^2:D_6$ 6A $6$ $6$ $C_6\wr C_2$ 3B 2A $d^{5}$
$C_6^2:D_6$ 6B $6$ $6$ $C_6\wr C_2$ 3A 2A $c^{5}$
$C_6^2:D_6$ 6C $6$ $12$ $C_6^2$ 3B 2A $c^{2}d$
$C_6^2:D_6$ 6D $6$ $36$ $C_2\times C_6$ 3B 2C $ab^{4}d^{2}$
$C_6^2:D_6$ 6E $6$ $36$ $C_2\times C_6$ 3A 2D $abd^{4}$
$C_6^2:D_6$ 6F $6$ $72$ $C_6$ 3C 2B $bc^{5}d^{3}$
$C_6^2:D_6$ 12A $12$ $36$ $C_{12}$ 6A 4A $ac^{5}d^{4}$
$C_6^2:D_6$ 12B $12$ $36$ $C_{12}$ 6B 4B $abcd^{3}$
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