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

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_3^3:S_3^2:C_2^2$ 1A $1$ $1$ $C_3^3:S_3^2:C_2^2$ 1A 1A $1$
$C_3^3:S_3^2:C_2^2$ 2A $2$ $18$ $C_6.S_3^2$ 1A 2A $acdg^{2}$
$C_3^3:S_3^2:C_2^2$ 2B $2$ $18$ $C_6.S_3^2$ 1A 2B $ab^{3}ceg$
$C_3^3:S_3^2:C_2^2$ 2C $2$ $27$ $S_3^2:C_2^2$ 1A 2C $b^{2}cd^{2}f$
$C_3^3:S_3^2:C_2^2$ 2D $2$ $27$ $S_3^2:C_2^2$ 1A 2D $c^{3}d^{2}f^{2}g^{2}$
$C_3^3:S_3^2:C_2^2$ 2E $2$ $54$ $S_3\times D_6$ 1A 2E $ac^{2}g^{2}$
$C_3^3:S_3^2:C_2^2$ 2F $2$ $54$ $S_3\times D_6$ 1A 2F $ab^{3}c^{4}efg$
$C_3^3:S_3^2:C_2^2$ 2G $2$ $81$ $S_3\times D_4$ 1A 2G $b^{2}c^{2}eg$
$C_3^3:S_3^2:C_2^2$ 3A $3$ $2$ $C_3^3:\SOPlus(4,2)$ 3A 1A $g^{2}$
$C_3^3:S_3^2:C_2^2$ 3B $3$ $12$ $C_3^3:D_6$ 3B 1A $d^{2}f$
$C_3^3:S_3^2:C_2^2$ 3C $3$ $12$ $C_3^3:D_6$ 3C 1A $d^{2}f^{2}g^{2}$
$C_3^3:S_3^2:C_2^2$ 3D $3$ $12$ $C_3^3:D_6$ 3D 1A $c^{4}$
$C_3^3:S_3^2:C_2^2$ 3E $3$ $12$ $C_3^3:D_6$ 3E 1A $c^{4}e^{2}g^{2}$
$C_3^3:S_3^2:C_2^2$ 3F $3$ $24$ $C_3^3:C_6$ 3F 1A $e^{2}f^{2}g$
$C_3^3:S_3^2:C_2^2$ 3G $3$ $24$ $C_3^3:S_3$ 3G 1A $d^{2}ef^{2}g^{2}$
$C_3^3:S_3^2:C_2^2$ 3H $3$ $24$ $C_3^3:C_6$ 3H 1A $c^{4}d^{2}e$
$C_3^3:S_3^2:C_2^2$ 3I $3$ $24$ $C_3^3:S_3$ 3I 1A $c^{2}d^{2}e$
$C_3^3:S_3^2:C_2^2$ 3J $3$ $48$ $C_3\times \He_3$ 3J 1A $de$
$C_3^3:S_3^2:C_2^2$ 3K $3$ $48$ $C_3\times \He_3$ 3K 1A $c^{2}ef$
$C_3^3:S_3^2:C_2^2$ 4A $4$ $162$ $C_4\times S_3$ 2G 4A $beg$
$C_3^3:S_3^2:C_2^2$ 4B $4$ $486$ $C_2\times C_4$ 2G 4B $bc^{3}eg^{2}$
$C_3^3:S_3^2:C_2^2$ 6A $6$ $36$ $C_3^2:D_6$ 3A 2A $acd$
$C_3^3:S_3^2:C_2^2$ 6B $6$ $36$ $C_3^2:D_6$ 3A 2B $ab^{3}ce$
$C_3^3:S_3^2:C_2^2$ 6C $6$ $108$ $C_6\times S_3$ 3B 2C $b^{2}c$
$C_3^3:S_3^2:C_2^2$ 6D $6$ $108$ $C_6\times S_3$ 3C 2C $b^{2}c^{5}ef^{2}g$
$C_3^3:S_3^2:C_2^2$ 6E $6$ $108$ $C_6\times S_3$ 3D 2D $c^{5}d^{2}f^{2}g^{2}$
$C_3^3:S_3^2:C_2^2$ 6F $6$ $108$ $C_6\times S_3$ 3C 2E $ab^{2}c^{2}eg$
$C_3^3:S_3^2:C_2^2$ 6G $6$ $108$ $C_6\times S_3$ 3B 2B $abc^{5}df^{2}g$
$C_3^3:S_3^2:C_2^2$ 6H $6$ $108$ $C_6\times S_3$ 3C 2A $acfg$
$C_3^3:S_3^2:C_2^2$ 6I $6$ $108$ $C_6\times S_3$ 3D 2A $acdef^{2}$
$C_3^3:S_3^2:C_2^2$ 6J $6$ $108$ $C_6\times S_3$ 3B 2F $abc^{2}d^{2}fg$
$C_3^3:S_3^2:C_2^2$ 6K $6$ $108$ $C_6\times S_3$ 3D 2E $ab^{2}d^{2}f^{2}$
$C_3^3:S_3^2:C_2^2$ 6L $6$ $108$ $C_6\times S_3$ 3E 2B $abc^{5}e^{2}fg^{2}$
$C_3^3:S_3^2:C_2^2$ 6M $6$ $108$ $C_6\times S_3$ 3E 2D $c^{5}de^{2}f$
$C_3^3:S_3^2:C_2^2$ 6N $6$ $108$ $C_6\times S_3$ 3E 2F $ab^{3}c^{2}d^{2}e^{2}f^{2}g$
$C_3^3:S_3^2:C_2^2$ 6O $6$ $162$ $C_3\times D_4$ 3A 2G $b^{2}c^{2}e$
$C_3^3:S_3^2:C_2^2$ 6P $6$ $216$ $C_3\times C_6$ 3F 2E $ac^{2}efg$
$C_3^3:S_3^2:C_2^2$ 6Q $6$ $216$ $C_3\times C_6$ 3G 2A $acd^{2}e^{2}g^{2}$
$C_3^3:S_3^2:C_2^2$ 6R $6$ $216$ $C_3\times C_6$ 3H 2F $ab^{3}dfg$
$C_3^3:S_3^2:C_2^2$ 6S $6$ $216$ $C_3\times C_6$ 3I 2B $abcf^{2}g$
$C_3^3:S_3^2:C_2^2$ 12A $12$ $324$ $C_{12}$ 6O 4A $b^{3}c^{2}g^{2}$
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