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

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_6^3:C_2$ 1A $1$ $1$ $C_6^3:C_2$ 1A 1A $1$
$C_6^3:C_2$ 2A $2$ $1$ $C_6^3:C_2$ 1A 2A $a$
$C_6^3:C_2$ 2B $2$ $1$ $C_6^3:C_2$ 1A 2B $d^{3}$
$C_6^3:C_2$ 2C $2$ $1$ $C_6^3:C_2$ 1A 2C $c^{3}$
$C_6^3:C_2$ 2D $2$ $1$ $C_6^3:C_2$ 1A 2D $ad^{3}$
$C_6^3:C_2$ 2E $2$ $1$ $C_6^3:C_2$ 1A 2E $ac^{3}$
$C_6^3:C_2$ 2F $2$ $1$ $C_6^3:C_2$ 1A 2F $c^{3}d^{3}$
$C_6^3:C_2$ 2G $2$ $1$ $C_6^3:C_2$ 1A 2G $ac^{3}d^{3}$
$C_6^3:C_2$ 2H $2$ $9$ $C_2^3\times C_6$ 1A 2H $b^{3}$
$C_6^3:C_2$ 2I $2$ $9$ $C_2^3\times C_6$ 1A 2I $b^{3}d$
$C_6^3:C_2$ 2J $2$ $9$ $C_2^3\times C_6$ 1A 2J $b^{3}c$
$C_6^3:C_2$ 2K $2$ $9$ $C_2^3\times C_6$ 1A 2K $ab^{3}$
$C_6^3:C_2$ 2L $2$ $9$ $C_2^3\times C_6$ 1A 2L $b^{3}cd$
$C_6^3:C_2$ 2M $2$ $9$ $C_2^3\times C_6$ 1A 2M $ab^{3}d$
$C_6^3:C_2$ 2N $2$ $9$ $C_2^3\times C_6$ 1A 2N $ab^{3}c$
$C_6^3:C_2$ 2O $2$ $9$ $C_2^3\times C_6$ 1A 2O $ab^{3}cd$
$C_6^3:C_2$ 3A1 $3$ $1$ $C_6^3:C_2$ 3A-1 1A $b^{4}$
$C_6^3:C_2$ 3A-1 $3$ $1$ $C_6^3:C_2$ 3A1 1A $b^{2}$
$C_6^3:C_2$ 3B $3$ $2$ $C_6^3$ 3B 1A $d^{2}$
$C_6^3:C_2$ 3C $3$ $2$ $C_6^3$ 3C 1A $c^{2}$
$C_6^3:C_2$ 3D $3$ $2$ $C_6^3$ 3D 1A $c^{4}d^{2}$
$C_6^3:C_2$ 3E $3$ $2$ $C_6^3$ 3E 1A $c^{2}d^{2}$
$C_6^3:C_2$ 3F1 $3$ $2$ $C_6^3$ 3F-1 1A $b^{4}d^{2}$
$C_6^3:C_2$ 3F-1 $3$ $2$ $C_6^3$ 3F1 1A $b^{2}d^{4}$
$C_6^3:C_2$ 3G1 $3$ $2$ $C_6^3$ 3G-1 1A $b^{4}c^{2}$
$C_6^3:C_2$ 3G-1 $3$ $2$ $C_6^3$ 3G1 1A $b^{2}c^{4}$
$C_6^3:C_2$ 3H1 $3$ $2$ $C_6^3$ 3H-1 1A $b^{4}c^{4}d^{2}$
$C_6^3:C_2$ 3H-1 $3$ $2$ $C_6^3$ 3H1 1A $b^{2}c^{2}d^{4}$
$C_6^3:C_2$ 3I1 $3$ $2$ $C_6^3$ 3I-1 1A $b^{4}c^{2}d^{2}$
$C_6^3:C_2$ 3I-1 $3$ $2$ $C_6^3$ 3I1 1A $b^{2}c^{4}d^{4}$
$C_6^3:C_2$ 6A1 $6$ $1$ $C_6^3:C_2$ 3A1 2A $ab^{2}$
$C_6^3:C_2$ 6A-1 $6$ $1$ $C_6^3:C_2$ 3A-1 2A $ab^{4}$
$C_6^3:C_2$ 6B1 $6$ $1$ $C_6^3:C_2$ 3A1 2B $b^{2}d^{3}$
$C_6^3:C_2$ 6B-1 $6$ $1$ $C_6^3:C_2$ 3A-1 2B $b^{4}d^{3}$
$C_6^3:C_2$ 6C1 $6$ $1$ $C_6^3:C_2$ 3A1 2C $b^{2}c^{3}$
$C_6^3:C_2$ 6C-1 $6$ $1$ $C_6^3:C_2$ 3A-1 2C $b^{4}c^{3}$
$C_6^3:C_2$ 6D1 $6$ $1$ $C_6^3:C_2$ 3A1 2D $ab^{2}d^{3}$
$C_6^3:C_2$ 6D-1 $6$ $1$ $C_6^3:C_2$ 3A-1 2D $ab^{4}d^{3}$
$C_6^3:C_2$ 6E1 $6$ $1$ $C_6^3:C_2$ 3A1 2E $ab^{2}c^{3}$
$C_6^3:C_2$ 6E-1 $6$ $1$ $C_6^3:C_2$ 3A-1 2E $ab^{4}c^{3}$
$C_6^3:C_2$ 6F1 $6$ $1$ $C_6^3:C_2$ 3A1 2F $b^{2}c^{3}d^{3}$
$C_6^3:C_2$ 6F-1 $6$ $1$ $C_6^3:C_2$ 3A-1 2F $b^{4}c^{3}d^{3}$
$C_6^3:C_2$ 6G1 $6$ $1$ $C_6^3:C_2$ 3A1 2G $ab^{2}c^{3}d^{3}$
$C_6^3:C_2$ 6G-1 $6$ $1$ $C_6^3:C_2$ 3A-1 2G $ab^{4}c^{3}d^{3}$
$C_6^3:C_2$ 6H $6$ $2$ $C_6^3$ 3B 2B $d$
$C_6^3:C_2$ 6I $6$ $2$ $C_6^3$ 3C 2C $c$
$C_6^3:C_2$ 6J $6$ $2$ $C_6^3$ 3D 2B $c^{2}d$
$C_6^3:C_2$ 6K $6$ $2$ $C_6^3$ 3D 2C $cd^{2}$
$C_6^3:C_2$ 6L $6$ $2$ $C_6^3$ 3E 2F $cd$
$C_6^3:C_2$ 6M $6$ $2$ $C_6^3$ 3B 2A $ad^{2}$
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