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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{6}=b^{3}=c^{3}=d^{9}=[b,c]=[b,d]=[c,d]=1, b^{a}=b^{2}, c^{a}=c^{2}d^{6}, d^{a}=cd^{8} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$(C_3^2\times C_9):C_6$ 1A $1$ $1$ $(C_3^2\times C_9):C_6$ 1A 1A $1$
$(C_3^2\times C_9):C_6$ 2A $2$ $81$ $C_6$ 1A 2A $a^{3}bcd^{6}$
$(C_3^2\times C_9):C_6$ 3A $3$ $2$ $\He_3:C_3^2$ 3A 1A $b$
$(C_3^2\times C_9):C_6$ 3B $3$ $2$ $\He_3:C_3^2$ 3B 1A $bd^{3}$
$(C_3^2\times C_9):C_6$ 3C $3$ $2$ $\He_3:C_3^2$ 3C 1A $bd^{6}$
$(C_3^2\times C_9):C_6$ 3D $3$ $2$ $\He_3:C_3^2$ 3D 1A $d^{3}$
$(C_3^2\times C_9):C_6$ 3E $3$ $6$ $C_3^2\times C_9$ 3E 1A $c$
$(C_3^2\times C_9):C_6$ 3F $3$ $6$ $C_3^2\times C_9$ 3F 1A $bc$
$(C_3^2\times C_9):C_6$ 3G $3$ $6$ $C_3^2\times C_9$ 3G 1A $bc^{2}$
$(C_3^2\times C_9):C_6$ 3H1 $3$ $9$ $C_3^2:C_6$ 3H-1 1A $a^{4}d^{3}$
$(C_3^2\times C_9):C_6$ 3H-1 $3$ $9$ $C_3^2:C_6$ 3H1 1A $a^{2}d^{6}$
$(C_3^2\times C_9):C_6$ 3I1 $3$ $18$ $C_3^3$ 3I-1 1A $a^{2}d$
$(C_3^2\times C_9):C_6$ 3I-1 $3$ $18$ $C_3^3$ 3I1 1A $a^{4}d$
$(C_3^2\times C_9):C_6$ 3J1 $3$ $18$ $C_3^3$ 3J-1 1A $a^{2}b$
$(C_3^2\times C_9):C_6$ 3J-1 $3$ $18$ $C_3^3$ 3J1 1A $a^{4}b$
$(C_3^2\times C_9):C_6$ 3K1 $3$ $18$ $C_3^3$ 3K-1 1A $a^{2}bd$
$(C_3^2\times C_9):C_6$ 3K-1 $3$ $18$ $C_3^3$ 3K1 1A $a^{4}bd$
$(C_3^2\times C_9):C_6$ 3L1 $3$ $18$ $C_3^3$ 3L-1 1A $a^{2}bd^{2}$
$(C_3^2\times C_9):C_6$ 3L-1 $3$ $18$ $C_3^3$ 3L1 1A $a^{4}bd^{2}$
$(C_3^2\times C_9):C_6$ 6A1 $6$ $81$ $C_6$ 3H1 2A $a^{5}bc$
$(C_3^2\times C_9):C_6$ 6A-1 $6$ $81$ $C_6$ 3H-1 2A $abcd^{3}$
$(C_3^2\times C_9):C_6$ 9A1 $9$ $6$ $C_3^2\times C_9$ 9A2 3D $d$
$(C_3^2\times C_9):C_6$ 9A2 $9$ $6$ $C_3^2\times C_9$ 9A4 3D $d^{2}$
$(C_3^2\times C_9):C_6$ 9A4 $9$ $6$ $C_3^2\times C_9$ 9A1 3D $d^{4}$
$(C_3^2\times C_9):C_6$ 9B1 $9$ $6$ $C_3^2\times C_9$ 9B2 3D $bd$
$(C_3^2\times C_9):C_6$ 9B2 $9$ $6$ $C_3^2\times C_9$ 9B4 3D $b^{2}d^{2}$
$(C_3^2\times C_9):C_6$ 9B4 $9$ $6$ $C_3^2\times C_9$ 9B1 3D $bd^{4}$
$(C_3^2\times C_9):C_6$ 9C1 $9$ $6$ $C_3^2\times C_9$ 9C2 3D $bd^{2}$
$(C_3^2\times C_9):C_6$ 9C2 $9$ $6$ $C_3^2\times C_9$ 9C4 3D $bd^{5}$
$(C_3^2\times C_9):C_6$ 9C4 $9$ $6$ $C_3^2\times C_9$ 9C1 3D $b^{2}d$
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