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

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_6.S_4$ 1A $1$ $1$ $C_6.S_4$ 1A 1A $1$
$C_6.S_4$ 2A $2$ $1$ $C_6.S_4$ 1A 2A $a^{2}$
$C_6.S_4$ 2B $2$ $3$ $C_6.D_4$ 1A 2B $d^{3}$
$C_6.S_4$ 2C $2$ $3$ $C_6.D_4$ 1A 2C $a^{2}d^{3}$
$C_6.S_4$ 3A $3$ $2$ $C_6\times A_4$ 3A 1A $d^{4}$
$C_6.S_4$ 3B $3$ $8$ $C_3\times C_6$ 3B 1A $b^{2}$
$C_6.S_4$ 3C $3$ $8$ $C_3\times C_6$ 3C 1A $b^{2}d^{4}$
$C_6.S_4$ 3D $3$ $8$ $C_3\times C_6$ 3D 1A $b^{2}cd^{5}$
$C_6.S_4$ 4A1 $4$ $18$ $C_2\times C_4$ 2C 4A-1 $abcd^{5}$
$C_6.S_4$ 4A-1 $4$ $18$ $C_2\times C_4$ 2C 4A1 $a^{3}bcd^{2}$
$C_6.S_4$ 4B1 $4$ $18$ $C_2\times C_4$ 2A 4B-1 $a^{3}b$
$C_6.S_4$ 4B-1 $4$ $18$ $C_2\times C_4$ 2A 4B1 $ab$
$C_6.S_4$ 6A $6$ $2$ $C_6\times A_4$ 3A 2A $a^{2}d^{2}$
$C_6.S_4$ 6B $6$ $6$ $C_2^2\times C_6$ 3A 2B $d$
$C_6.S_4$ 6C $6$ $6$ $C_2^2\times C_6$ 3A 2C $a^{2}d$
$C_6.S_4$ 6D $6$ $8$ $C_3\times C_6$ 3B 2A $a^{2}b$
$C_6.S_4$ 6E $6$ $8$ $C_3\times C_6$ 3C 2A $a^{2}bd^{2}$
$C_6.S_4$ 6F $6$ $8$ $C_3\times C_6$ 3D 2A $a^{2}bd$
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