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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{6}=b^{6}=c^{12}=d^{3}=[a,c]=[a,d]=[b,d]=1, b^{a}=b^{5}, c^{b}=c^{11}, d^{c}=d^{2} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_6.S_3^3$ 1A $1$ $1$ $C_6.S_3^3$ 1A 1A $1$
$C_6.S_3^3$ 2A $2$ $1$ $C_6.S_3^3$ 1A 2A $c^{6}$
$C_6.S_3^3$ 2B $2$ $3$ $C_6^2.D_6$ 1A 2B $a^{3}$
$C_6.S_3^3$ 2C $2$ $3$ $C_6^2.D_6$ 1A 2C $a^{3}c^{6}$
$C_6.S_3^3$ 2D $2$ $6$ $S_3\times C_6^2$ 1A 2D $b^{3}$
$C_6.S_3^3$ 2E $2$ $18$ $C_2\times C_6^2$ 1A 2E $a^{3}b$
$C_6.S_3^3$ 2F $2$ $18$ $C_6\times D_6$ 1A 2F $b^{3}c$
$C_6.S_3^3$ 2G $2$ $54$ $C_2^2\times C_6$ 1A 2G $a^{3}bc$
$C_6.S_3^3$ 3A1 $3$ $1$ $C_6.S_3^3$ 3A-1 1A $a^{2}$
$C_6.S_3^3$ 3A-1 $3$ $1$ $C_6.S_3^3$ 3A1 1A $a^{4}$
$C_6.S_3^3$ 3B $3$ $2$ $C_3^4:C_2^3$ 3B 1A $d^{2}$
$C_6.S_3^3$ 3C $3$ $2$ $S_3\times C_3^2:C_{12}$ 3C 1A $c^{8}$
$C_6.S_3^3$ 3D $3$ $2$ $C_3^3:D_{12}$ 3D 1A $b^{2}$
$C_6.S_3^3$ 3E1 $3$ $2$ $C_3^4:C_2^3$ 3E-1 1A $a^{4}d^{2}$
$C_6.S_3^3$ 3E-1 $3$ $2$ $C_3^4:C_2^3$ 3E1 1A $a^{2}d$
$C_6.S_3^3$ 3F1 $3$ $2$ $S_3\times C_3^2:C_{12}$ 3F-1 1A $a^{2}c^{8}$
$C_6.S_3^3$ 3F-1 $3$ $2$ $S_3\times C_3^2:C_{12}$ 3F1 1A $a^{4}c^{4}$
$C_6.S_3^3$ 3G1 $3$ $2$ $C_3^3:D_{12}$ 3G-1 1A $a^{4}b^{2}$
$C_6.S_3^3$ 3G-1 $3$ $2$ $C_3^3:D_{12}$ 3G1 1A $a^{2}b^{4}$
$C_6.S_3^3$ 3H $3$ $4$ $D_6\times C_3^3$ 3H 1A $c^{4}d^{2}$
$C_6.S_3^3$ 3I $3$ $4$ $D_6\times C_3^3$ 3I 1A $b^{4}d^{2}$
$C_6.S_3^3$ 3J $3$ $4$ $C_3^3:C_{12}$ 3J 1A $b^{2}c^{4}$
$C_6.S_3^3$ 3K1 $3$ $4$ $D_6\times C_3^3$ 3K-1 1A $a^{4}c^{4}d^{2}$
$C_6.S_3^3$ 3K-1 $3$ $4$ $D_6\times C_3^3$ 3K1 1A $a^{2}c^{8}d$
$C_6.S_3^3$ 3L1 $3$ $4$ $D_6\times C_3^3$ 3L-1 1A $a^{4}b^{4}d^{2}$
$C_6.S_3^3$ 3L-1 $3$ $4$ $D_6\times C_3^3$ 3L1 1A $a^{2}b^{2}d$
$C_6.S_3^3$ 3M1 $3$ $4$ $C_3^3:C_{12}$ 3M-1 1A $a^{2}b^{2}c^{4}$
$C_6.S_3^3$ 3M-1 $3$ $4$ $C_3^3:C_{12}$ 3M1 1A $a^{4}b^{4}c^{8}$
$C_6.S_3^3$ 3N $3$ $8$ $C_3^3\times C_6$ 3N 1A $b^{4}c^{4}d^{2}$
$C_6.S_3^3$ 3O1 $3$ $8$ $C_3^3\times C_6$ 3O-1 1A $a^{4}b^{4}c^{4}d^{2}$
$C_6.S_3^3$ 3O-1 $3$ $8$ $C_3^3\times C_6$ 3O1 1A $a^{2}b^{2}c^{8}d$
$C_6.S_3^3$ 4A $4$ $6$ $C_6.C_6^2$ 2A 4A $c^{3}$
$C_6.S_3^3$ 4B $4$ $18$ $C_6\times C_{12}$ 2A 4B $a^{3}c^{9}$
$C_6.S_3^3$ 6A1 $6$ $1$ $C_6.S_3^3$ 3A1 2A $a^{4}c^{6}$
$C_6.S_3^3$ 6A-1 $6$ $1$ $C_6.S_3^3$ 3A-1 2A $a^{2}c^{6}$
$C_6.S_3^3$ 6B $6$ $2$ $C_3^4:C_2^3$ 3B 2A $c^{6}d$
$C_6.S_3^3$ 6C $6$ $2$ $S_3\times C_3^2:C_{12}$ 3C 2A $c^{2}$
$C_6.S_3^3$ 6D $6$ $2$ $C_3^3:D_{12}$ 3D 2A $b^{4}c^{6}$
$C_6.S_3^3$ 6E1 $6$ $2$ $C_3^4:C_2^3$ 3E1 2A $a^{2}c^{6}d$
$C_6.S_3^3$ 6E-1 $6$ $2$ $C_3^4:C_2^3$ 3E-1 2A $a^{4}c^{6}d$
$C_6.S_3^3$ 6F1 $6$ $2$ $S_3\times C_3^2:C_{12}$ 3F1 2A $a^{4}c^{2}$
$C_6.S_3^3$ 6F-1 $6$ $2$ $S_3\times C_3^2:C_{12}$ 3F-1 2A $a^{2}c^{10}$
$C_6.S_3^3$ 6G1 $6$ $2$ $C_3^3:D_{12}$ 3G-1 2A $a^{4}b^{4}c^{6}$
$C_6.S_3^3$ 6G-1 $6$ $2$ $C_3^3:D_{12}$ 3G1 2A $a^{2}b^{2}c^{6}$
$C_6.S_3^3$ 6H1 $6$ $3$ $C_6^2.D_6$ 3A1 2B $a$
$C_6.S_3^3$ 6H-1 $6$ $3$ $C_6^2.D_6$ 3A-1 2B $a^{5}$
$C_6.S_3^3$ 6I1 $6$ $3$ $C_6^2.D_6$ 3A1 2C $ac^{6}$
$C_6.S_3^3$ 6I-1 $6$ $3$ $C_6^2.D_6$ 3A-1 2C $a^{5}c^{6}$
$C_6.S_3^3$ 6J $6$ $4$ $D_6\times C_3^3$ 3H 2A $c^{2}d$
$C_6.S_3^3$ 6K $6$ $4$ $D_6\times C_3^3$ 3I 2A $b^{2}c^{6}d$
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