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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{6}=b^{18}=c^{6}=d^{3}=[a,c]=[a,d]=[b,d]=1, b^{a}=b^{11}, c^{b}=c^{5}, d^{c}=d^{2} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_3^2.S_3^3$ 1A $1$ $1$ $C_3^2.S_3^3$ 1A 1A $1$
$C_3^2.S_3^3$ 2A $2$ $3$ $C_{18}:C_6\times S_3$ 1A 2A $c^{3}$
$C_3^2.S_3^3$ 2B $2$ $3$ $C_{18}:C_6\times S_3$ 1A 2B $b^{9}$
$C_3^2.S_3^3$ 2C $2$ $9$ $C_6\times S_3^2$ 1A 2C $a^{3}$
$C_3^2.S_3^3$ 2D $2$ $9$ $D_{18}:C_6$ 1A 2D $b^{9}c$
$C_3^2.S_3^3$ 2E $2$ $27$ $C_6\times D_6$ 1A 2E $a^{3}b^{7}$
$C_3^2.S_3^3$ 2F $2$ $27$ $C_6\times D_6$ 1A 2F $a^{3}c^{3}$
$C_3^2.S_3^3$ 2G $2$ $81$ $C_2^2\times C_6$ 1A 2G $a^{3}b^{11}c^{3}d^{2}$
$C_3^2.S_3^3$ 3A $3$ $2$ $C_3^3.S_3^2$ 3A 1A $c^{2}$
$C_3^2.S_3^3$ 3B $3$ $2$ $C_3^3.S_3^2$ 3B 1A $d^{2}$
$C_3^2.S_3^3$ 3C $3$ $2$ $C_3^3.C_6^2$ 3C 1A $b^{6}$
$C_3^2.S_3^3$ 3D1 $3$ $3$ $C_3\times S_3^3$ 3D-1 1A $a^{2}$
$C_3^2.S_3^3$ 3D-1 $3$ $3$ $C_3\times S_3^3$ 3D1 1A $a^{4}$
$C_3^2.S_3^3$ 3E $3$ $4$ $C_3^4.C_6$ 3E 1A $b^{12}c^{2}$
$C_3^2.S_3^3$ 3F $3$ $4$ $C_3^4.C_6$ 3F 1A $b^{6}d^{2}$
$C_3^2.S_3^3$ 3G $3$ $4$ $D_9:C_3^3$ 3G 1A $c^{4}d^{2}$
$C_3^2.S_3^3$ 3H1 $3$ $6$ $C_3^2\times S_3^2$ 3H-1 1A $a^{4}c^{2}$
$C_3^2.S_3^3$ 3H-1 $3$ $6$ $C_3^2\times S_3^2$ 3H1 1A $a^{2}c^{4}$
$C_3^2.S_3^3$ 3I1 $3$ $6$ $C_3^2\times S_3^2$ 3I-1 1A $a^{2}d^{2}$
$C_3^2.S_3^3$ 3I-1 $3$ $6$ $C_3^2\times S_3^2$ 3I1 1A $a^{4}d$
$C_3^2.S_3^3$ 3J $3$ $8$ $C_9:C_3^3$ 3J 1A $b^{6}c^{2}d$
$C_3^2.S_3^3$ 3K1 $3$ $12$ $S_3\times C_3^3$ 3K-1 1A $a^{2}c^{4}d^{2}$
$C_3^2.S_3^3$ 3K-1 $3$ $12$ $S_3\times C_3^3$ 3K1 1A $a^{4}c^{2}d$
$C_3^2.S_3^3$ 6A $6$ $6$ $C_9:C_6^2$ 3A 2A $c$
$C_3^2.S_3^3$ 6B $6$ $6$ $C_9:C_6^2$ 3B 2B $b^{9}d$
$C_3^2.S_3^3$ 6C $6$ $6$ $C_3^2.C_6^2$ 3C 2B $b^{3}$
$C_3^2.S_3^3$ 6D $6$ $6$ $C_3^2.C_6^2$ 3C 2A $b^{6}c^{3}$
$C_3^2.S_3^3$ 6E1 $6$ $9$ $C_6\times S_3^2$ 3D1 2C $a$
$C_3^2.S_3^3$ 6E-1 $6$ $9$ $C_6\times S_3^2$ 3D-1 2C $a^{5}$
$C_3^2.S_3^3$ 6F1 $6$ $9$ $C_6\times S_3^2$ 3D-1 2A $a^{2}c^{3}$
$C_3^2.S_3^3$ 6F-1 $6$ $9$ $C_6\times S_3^2$ 3D1 2A $a^{4}c^{3}$
$C_3^2.S_3^3$ 6G1 $6$ $9$ $C_6\times S_3^2$ 3D-1 2B $a^{2}b^{3}$
$C_3^2.S_3^3$ 6G-1 $6$ $9$ $C_6\times S_3^2$ 3D1 2B $a^{4}b^{3}$
$C_3^2.S_3^3$ 6H $6$ $12$ $C_{18}:C_3^2$ 3E 2A $b^{6}c$
$C_3^2.S_3^3$ 6I $6$ $12$ $C_{18}:C_3^2$ 3F 2B $b^{3}d$
$C_3^2.S_3^3$ 6J $6$ $18$ $C_3^2\times D_6$ 3B 2C $a^{3}d$
$C_3^2.S_3^3$ 6K $6$ $18$ $C_3^2\times D_6$ 3A 2C $a^{3}c^{2}$
$C_3^2.S_3^3$ 6L $6$ $18$ $C_{18}:C_6$ 3C 2D $b^{3}c^{5}$
$C_3^2.S_3^3$ 6M1 $6$ $18$ $C_3^2\times D_6$ 3H1 2A $a^{2}c$
$C_3^2.S_3^3$ 6M-1 $6$ $18$ $C_3^2\times D_6$ 3H-1 2A $a^{4}c$
$C_3^2.S_3^3$ 6N1 $6$ $18$ $C_3^2\times D_6$ 3I1 2C $ad$
$C_3^2.S_3^3$ 6N-1 $6$ $18$ $C_3^2\times D_6$ 3I-1 2C $a^{5}d$
$C_3^2.S_3^3$ 6O1 $6$ $18$ $C_3^2\times D_6$ 3H-1 2C $ac^{2}$
$C_3^2.S_3^3$ 6O-1 $6$ $18$ $C_3^2\times D_6$ 3H1 2C $a^{5}c^{2}$
$C_3^2.S_3^3$ 6P1 $6$ $18$ $C_3^2\times D_6$ 3I-1 2B $a^{2}b^{3}d$
$C_3^2.S_3^3$ 6P-1 $6$ $18$ $C_3^2\times D_6$ 3I1 2B $a^{4}b^{3}d$
$C_3^2.S_3^3$ 6Q1 $6$ $27$ $C_6\times D_6$ 3D1 2E $ab$
$C_3^2.S_3^3$ 6Q-1 $6$ $27$ $C_6\times D_6$ 3D-1 2E $a^{5}b$
$C_3^2.S_3^3$ 6R1 $6$ $27$ $C_6\times D_6$ 3D1 2F $ac^{3}$
$C_3^2.S_3^3$ 6R-1 $6$ $27$ $C_6\times D_6$ 3D-1 2F $a^{5}c^{3}$
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