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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{6}=b^{18}=c^{2}=d^{6}=[a,d]=[c,d]=1, b^{a}=b^{5}, c^{a}=cd^{3}, c^{b}=cd^{3}, d^{b}=cd^{2} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_6^2.S_3^2$ 1A $1$ $1$ $C_6^2.S_3^2$ 1A 1A $1$
$C_6^2.S_3^2$ 2A $2$ $3$ $C_6^2:D_6$ 1A 2A $c$
$C_6^2.S_3^2$ 2B $2$ $3$ $C_6^2.D_6$ 1A 2B $b^{9}$
$C_6^2.S_3^2$ 2C $2$ $9$ $C_6^2:C_2^2$ 1A 2C $b^{9}c$
$C_6^2.S_3^2$ 2D $2$ $18$ $C_6\times D_6$ 1A 2D $a^{3}$
$C_6^2.S_3^2$ 2E $2$ $54$ $C_2^2\times C_6$ 1A 2E $a^{3}b^{13}d^{4}$
$C_6^2.S_3^2$ 3A $3$ $2$ $S_3\times C_3^2.A_4$ 3A 1A $b^{12}$
$C_6^2.S_3^2$ 3B $3$ $2$ $C_3^3.S_4$ 3B 1A $d^{4}$
$C_6^2.S_3^2$ 3C1 $3$ $3$ $C_6^2:D_6$ 3C-1 1A $a^{4}$
$C_6^2.S_3^2$ 3C-1 $3$ $3$ $C_6^2:D_6$ 3C1 1A $a^{2}$
$C_6^2.S_3^2$ 3D $3$ $4$ $C_3^3.A_4$ 3D 1A $b^{12}d^{2}$
$C_6^2.S_3^2$ 3E1 $3$ $6$ $C_6^2:C_6$ 3E-1 1A $a^{4}d^{4}$
$C_6^2.S_3^2$ 3E-1 $3$ $6$ $C_6^2:C_6$ 3E1 1A $a^{2}d^{2}$
$C_6^2.S_3^2$ 4A $4$ $18$ $S_3\times C_{12}$ 2A 4A $a^{3}cd^{3}$
$C_6^2.S_3^2$ 4B $4$ $54$ $C_2\times C_{12}$ 2A 4B $a^{3}b^{15}cd^{2}$
$C_6^2.S_3^2$ 6A1 $6$ $3$ $C_6^2:D_6$ 3C1 2A $a^{2}d^{3}$
$C_6^2.S_3^2$ 6A-1 $6$ $3$ $C_6^2:D_6$ 3C-1 2A $a^{4}d^{3}$
$C_6^2.S_3^2$ 6B $6$ $6$ $S_3\times C_6^2$ 3A 2A $b^{6}c$
$C_6^2.S_3^2$ 6C $6$ $6$ $C_6^2:C_6$ 3B 2A $d$
$C_6^2.S_3^2$ 6D $6$ $6$ $C_6^2.C_6$ 3A 2B $b^{3}$
$C_6^2.S_3^2$ 6E1 $6$ $6$ $S_3\times C_6^2$ 3C1 2A $a^{2}c$
$C_6^2.S_3^2$ 6E-1 $6$ $6$ $S_3\times C_6^2$ 3C-1 2A $a^{4}c$
$C_6^2.S_3^2$ 6F1 $6$ $6$ $C_6^2:C_6$ 3E1 2A $a^{2}d$
$C_6^2.S_3^2$ 6F-1 $6$ $6$ $C_6^2:C_6$ 3E-1 2A $a^{4}d^{5}$
$C_6^2.S_3^2$ 6G1 $6$ $9$ $C_6^2:C_2^2$ 3C1 2B $a^{2}b^{3}$
$C_6^2.S_3^2$ 6G-1 $6$ $9$ $C_6^2:C_2^2$ 3C-1 2B $a^{4}b^{3}$
$C_6^2.S_3^2$ 6H1 $6$ $9$ $C_6^2:C_2^2$ 3C1 2C $a^{2}b^{3}c$
$C_6^2.S_3^2$ 6H-1 $6$ $9$ $C_6^2:C_2^2$ 3C-1 2C $a^{4}b^{3}cd$
$C_6^2.S_3^2$ 6I $6$ $12$ $C_3\times C_6^2$ 3D 2A $b^{6}d$
$C_6^2.S_3^2$ 6J1 $6$ $12$ $C_3\times C_6^2$ 3E1 2A $a^{2}cd^{2}$
$C_6^2.S_3^2$ 6J-1 $6$ $12$ $C_3\times C_6^2$ 3E-1 2A $a^{4}cd^{2}$
$C_6^2.S_3^2$ 6K $6$ $18$ $C_2\times C_6^2$ 3A 2C $b^{3}d$
$C_6^2.S_3^2$ 6L1 $6$ $18$ $C_6\times D_6$ 3C-1 2D $a$
$C_6^2.S_3^2$ 6L-1 $6$ $18$ $C_6\times D_6$ 3C1 2D $a^{5}$
$C_6^2.S_3^2$ 6M1 $6$ $18$ $C_2\times C_6^2$ 3C1 2C $a^{2}b^{3}d$
$C_6^2.S_3^2$ 6M-1 $6$ $18$ $C_2\times C_6^2$ 3C-1 2C $a^{4}b^{3}d$
$C_6^2.S_3^2$ 6N $6$ $36$ $C_6^2$ 3B 2D $a^{3}d^{2}$
$C_6^2.S_3^2$ 6O1 $6$ $36$ $C_6^2$ 3E-1 2D $ad^{2}$
$C_6^2.S_3^2$ 6O-1 $6$ $36$ $C_6^2$ 3E1 2D $a^{5}d^{2}$
$C_6^2.S_3^2$ 6P1 $6$ $54$ $C_2^2\times C_6$ 3C-1 2E $abd^{4}$
$C_6^2.S_3^2$ 6P-1 $6$ $54$ $C_2^2\times C_6$ 3C1 2E $a^{5}b^{7}d^{4}$
$C_6^2.S_3^2$ 9A $9$ $24$ $S_3\times C_9$ 9A 3A $b^{14}$
$C_6^2.S_3^2$ 9B1 $9$ $24$ $S_3\times C_9$ 9B-1 3A $a^{4}b^{8}cd^{3}$
$C_6^2.S_3^2$ 9B-1 $9$ $24$ $S_3\times C_9$ 9B1 3A $a^{2}b^{16}d^{3}$
$C_6^2.S_3^2$ 9C $9$ $48$ $C_3\times C_9$ 9C 3A $b^{4}cd^{5}$
$C_6^2.S_3^2$ 9D1 $9$ $48$ $C_3\times C_9$ 9D-1 3A $a^{2}b^{16}d^{2}$
$C_6^2.S_3^2$ 9D-1 $9$ $48$ $C_3\times C_9$ 9D1 3A $a^{4}b^{8}d^{4}$
$C_6^2.S_3^2$ 12A1 $12$ $18$ $S_3\times C_{12}$ 6A1 4A $ac$
$C_6^2.S_3^2$ 12A-1 $12$ $18$ $S_3\times C_{12}$ 6A-1 4A $a^{5}c$
$C_6^2.S_3^2$ 12B $12$ $36$ $C_3\times C_{12}$ 6C 4A $a^{3}cd^{2}$
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