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Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e \mid b^{18}=c^{4}=d^{6}=e^{6}=[a,e]=[d,e]=1, a^{2}=e^{3}, b^{a}=b^{17}d^{4}e^{5}, c^{a}=cd^{4}, d^{a}=d^{5}e^{3}, c^{b}=c^{3}e^{2}, d^{b}=d^{4}e^{3}, e^{b}=d^{5}e, d^{c}=d^{5}, e^{c}=e^{5} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_6^2:(D_4\times D_9)$ 1A $1$ $1$ $C_6^2:(D_4\times D_9)$ 1A 1A $1$
$C_6^2:(D_4\times D_9)$ 2A $2$ $1$ $C_6^2:(D_4\times D_9)$ 1A 2A $c^{2}$
$C_6^2:(D_4\times D_9)$ 2B $2$ $2$ $C_2^2\times C_6^2.(C_3\times S_3)$ 1A 2B $b^{9}$
$C_6^2:(D_4\times D_9)$ 2C $2$ $3$ $C_3^3.D_4^2$ 1A 2C $c^{2}e^{3}$
$C_6^2:(D_4\times D_9)$ 2D $2$ $3$ $C_3^3.D_4^2$ 1A 2D $e^{3}$
$C_6^2:(D_4\times D_9)$ 2E $2$ $6$ $C_6^3:C_2^2$ 1A 2E $b^{9}e^{3}$
$C_6^2:(D_4\times D_9)$ 2F $2$ $18$ $C_2^3:D_{18}$ 1A 2F $b^{9}cd^{4}e^{2}$
$C_6^2:(D_4\times D_9)$ 2G $2$ $54$ $C_2^3:D_6$ 1A 2G $b^{9}c^{3}d^{5}e$
$C_6^2:(D_4\times D_9)$ 2H $2$ $54$ $C_2^3:D_6$ 1A 2H $ab^{6}d$
$C_6^2:(D_4\times D_9)$ 2I $2$ $54$ $C_2^3:D_6$ 1A 2I $ac^{2}d^{5}e^{3}$
$C_6^2:(D_4\times D_9)$ 2J $2$ $108$ $C_2^2\times D_6$ 1A 2J $ab^{5}cd^{5}e$
$C_6^2:(D_4\times D_9)$ 2K $2$ $108$ $C_2^3\times C_6$ 1A 2K $ab^{15}c^{2}d^{3}$
$C_6^2:(D_4\times D_9)$ 3A $3$ $2$ $C_3^3.\GL(2,\mathbb{Z}/4)$ 3A 1A $d^{4}$
$C_6^2:(D_4\times D_9)$ 3B $3$ $2$ $C_6^2:(D_4\times C_9)$ 3B 1A $b^{12}$
$C_6^2:(D_4\times D_9)$ 3C $3$ $4$ $C_6^3.C_6$ 3C 1A $b^{12}d^{4}$
$C_6^2:(D_4\times D_9)$ 3D $3$ $6$ $C_6^3:C_2^2$ 3D 1A $e^{4}$
$C_6^2:(D_4\times D_9)$ 3E $3$ $12$ $C_2\times C_6^3$ 3E 1A $b^{12}e^{2}$
$C_6^2:(D_4\times D_9)$ 4A $4$ $18$ $C_{12}.S_4$ 2A 4A $cd^{4}e^{2}$
$C_6^2:(D_4\times D_9)$ 4B $4$ $54$ $C_{12}:D_4$ 2D 4B $ab^{4}c^{2}d^{3}e^{2}$
$C_6^2:(D_4\times D_9)$ 4C $4$ $54$ $C_{12}:D_4$ 2D 4C $ab^{12}d^{4}$
$C_6^2:(D_4\times D_9)$ 4D $4$ $54$ $C_{12}:D_4$ 2A 4D $c^{3}d^{2}e^{3}$
$C_6^2:(D_4\times D_9)$ 4E $4$ $108$ $C_4\times D_6$ 2A 4E $ab^{4}cd^{5}e$
$C_6^2:(D_4\times D_9)$ 4F $4$ $108$ $C_2^2\times C_{12}$ 2D 4F $abd^{2}e^{2}$
$C_6^2:(D_4\times D_9)$ 4G $4$ $108$ $C_{12}:C_4$ 2C 4G $ab^{12}cd^{2}$
$C_6^2:(D_4\times D_9)$ 4H $4$ $108$ $C_4\times D_6$ 2D 4H $ab^{13}c^{3}d^{3}e^{2}$
$C_6^2:(D_4\times D_9)$ 6A $6$ $2$ $C_6^2:(D_4\times C_9)$ 3B 2A $b^{6}c^{2}$
$C_6^2:(D_4\times D_9)$ 6B $6$ $2$ $C_3^3.\GL(2,\mathbb{Z}/4)$ 3A 2A $c^{2}d^{4}$
$C_6^2:(D_4\times D_9)$ 6C $6$ $4$ $C_6^3.C_6$ 3C 2A $b^{6}c^{2}d^{2}$
$C_6^2:(D_4\times D_9)$ 6D $6$ $4$ $C_6^3.C_6$ 3C 2B $b^{3}d^{2}$
$C_6^2:(D_4\times D_9)$ 6E $6$ $4$ $C_6^3.C_6$ 3C 2B $b^{3}d^{4}$
$C_6^2:(D_4\times D_9)$ 6F $6$ $4$ $C_6^3.C_6$ 3A 2B $b^{9}d^{2}$
$C_6^2:(D_4\times D_9)$ 6G $6$ $4$ $C_6^3.C_6$ 3B 2B $b^{3}$
$C_6^2:(D_4\times D_9)$ 6H $6$ $6$ $C_6^3:C_2^2$ 3D 2A $c^{2}e^{2}$
$C_6^2:(D_4\times D_9)$ 6I $6$ $6$ $C_6^3:C_2^2$ 3D 2C $c^{2}e$
$C_6^2:(D_4\times D_9)$ 6J $6$ $6$ $C_6^3:C_2^2$ 3B 2D $b^{6}e^{3}$
$C_6^2:(D_4\times D_9)$ 6K $6$ $6$ $C_6^3:C_2^2$ 3B 2C $b^{6}c^{2}e^{3}$
$C_6^2:(D_4\times D_9)$ 6L $6$ $6$ $C_6^3:C_2^2$ 3D 2D $d^{5}e^{2}$
$C_6^2:(D_4\times D_9)$ 6M $6$ $6$ $C_6^3.C_2^2$ 3A 2C $c^{2}d^{2}e^{3}$
$C_6^2:(D_4\times D_9)$ 6N $6$ $6$ $C_6^3.C_2^2$ 3A 2D $d^{5}$
$C_6^2:(D_4\times D_9)$ 6O1 $6$ $6$ $C_6^3:C_2^2$ 3D 2B $b^{9}e^{2}$
$C_6^2:(D_4\times D_9)$ 6O-1 $6$ $6$ $C_6^3:C_2^2$ 3D 2B $b^{9}e^{4}$
$C_6^2:(D_4\times D_9)$ 6P1 $6$ $6$ $C_6^3:C_2^2$ 3D 2E $b^{9}e$
$C_6^2:(D_4\times D_9)$ 6P-1 $6$ $6$ $C_6^3:C_2^2$ 3D 2E $b^{9}c^{2}e$
$C_6^2:(D_4\times D_9)$ 6Q $6$ $12$ $C_2\times C_6^3$ 3D 2D $d^{2}e$
$C_6^2:(D_4\times D_9)$ 6R $6$ $12$ $C_2\times C_6^3$ 3E 2D $b^{6}e$
$C_6^2:(D_4\times D_9)$ 6S $6$ $12$ $C_2\times C_6^3$ 3C 2D $b^{6}d$
$C_6^2:(D_4\times D_9)$ 6T $6$ $12$ $C_2\times C_6^3$ 3D 2C $c^{2}d^{2}e$
$C_6^2:(D_4\times D_9)$ 6U $6$ $12$ $C_2\times C_6^3$ 3E 2A $b^{6}c^{2}e^{2}$
$C_6^2:(D_4\times D_9)$ 6V $6$ $12$ $C_2\times C_6^3$ 3E 2C $b^{6}c^{2}e$
$C_6^2:(D_4\times D_9)$ 6W $6$ $12$ $C_2\times C_6^3$ 3C 2C $b^{6}c^{2}d$
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