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Elements of the group are displayed as words in the presentation $\langle a, b, c \mid a^{6}=b^{9}=c^{18}=[b,c]=1, b^{a}=b^{2}, c^{a}=c^{5} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_2\times C_9^2:C_6$ 1A $1$ $1$ $C_2\times C_9^2:C_6$ 1A 1A $1$
$C_2\times C_9^2:C_6$ 2A $2$ $1$ $C_2\times C_9^2:C_6$ 1A 2A $c^{9}$
$C_2\times C_9^2:C_6$ 2B $2$ $81$ $C_2\times C_6$ 1A 2B $a^{3}c^{11}$
$C_2\times C_9^2:C_6$ 2C $2$ $81$ $C_2\times C_6$ 1A 2C $a^{3}c^{6}$
$C_2\times C_9^2:C_6$ 3A $3$ $2$ $C_9^2:C_6$ 3A 1A $c^{6}$
$C_2\times C_9^2:C_6$ 3B $3$ $2$ $C_9^2:C_6$ 3B 1A $b^{6}$
$C_2\times C_9^2:C_6$ 3C $3$ $2$ $C_9^2:C_6$ 3C 1A $b^{6}c^{6}$
$C_2\times C_9^2:C_6$ 3D $3$ $2$ $C_9^2:C_6$ 3D 1A $b^{3}c^{6}$
$C_2\times C_9^2:C_6$ 3E1 $3$ $9$ $C_3^2:D_6$ 3E-1 1A $a^{4}c^{6}$
$C_2\times C_9^2:C_6$ 3E-1 $3$ $9$ $C_3^2:D_6$ 3E1 1A $a^{2}c^{12}$
$C_2\times C_9^2:C_6$ 6A $6$ $2$ $C_9^2:C_6$ 3A 2A $c^{3}$
$C_2\times C_9^2:C_6$ 6B $6$ $2$ $C_9^2:C_6$ 3B 2A $b^{3}c^{9}$
$C_2\times C_9^2:C_6$ 6C $6$ $2$ $C_9^2:C_6$ 3C 2A $b^{3}c^{3}$
$C_2\times C_9^2:C_6$ 6D $6$ $2$ $C_9^2:C_6$ 3D 2A $b^{6}c^{3}$
$C_2\times C_9^2:C_6$ 6E1 $6$ $9$ $C_3^2:D_6$ 3E1 2A $a^{2}c^{3}$
$C_2\times C_9^2:C_6$ 6E-1 $6$ $9$ $C_3^2:D_6$ 3E-1 2A $a^{4}c^{3}$
$C_2\times C_9^2:C_6$ 6F1 $6$ $81$ $C_2\times C_6$ 3E-1 2B $ac^{5}$
$C_2\times C_9^2:C_6$ 6F-1 $6$ $81$ $C_2\times C_6$ 3E1 2B $a^{5}c^{17}$
$C_2\times C_9^2:C_6$ 6G1 $6$ $81$ $C_2\times C_6$ 3E-1 2C $ac^{6}$
$C_2\times C_9^2:C_6$ 6G-1 $6$ $81$ $C_2\times C_6$ 3E1 2C $a^{5}c^{6}$
$C_2\times C_9^2:C_6$ 9A $9$ $6$ $C_9\times C_{18}$ 9A 3A $c^{2}$
$C_2\times C_9^2:C_6$ 9B $9$ $6$ $C_9\times C_{18}$ 9B 3A $b^{6}c^{2}$
$C_2\times C_9^2:C_6$ 9C $9$ $6$ $C_9\times C_{18}$ 9C 3A $b^{3}c^{2}$
$C_2\times C_9^2:C_6$ 9D $9$ $6$ $C_9\times C_{18}$ 9D 3B $b^{2}c^{6}$
$C_2\times C_9^2:C_6$ 9E $9$ $6$ $C_9\times C_{18}$ 9E 3B $b^{4}c^{6}$
$C_2\times C_9^2:C_6$ 9F $9$ $6$ $C_9\times C_{18}$ 9F 3B $b^{2}$
$C_2\times C_9^2:C_6$ 9G1 $9$ $6$ $C_9\times C_{18}$ 9G2 3C $b^{2}c^{2}$
$C_2\times C_9^2:C_6$ 9G2 $9$ $6$ $C_9\times C_{18}$ 9G4 3C $b^{4}c^{4}$
$C_2\times C_9^2:C_6$ 9G4 $9$ $6$ $C_9\times C_{18}$ 9G1 3C $b^{8}c^{8}$
$C_2\times C_9^2:C_6$ 9H1 $9$ $6$ $C_9\times C_{18}$ 9H2 3D $b^{4}c^{2}$
$C_2\times C_9^2:C_6$ 9H2 $9$ $6$ $C_9\times C_{18}$ 9H4 3D $b^{8}c^{4}$
$C_2\times C_9^2:C_6$ 9H4 $9$ $6$ $C_9\times C_{18}$ 9H1 3D $b^{7}c^{8}$
$C_2\times C_9^2:C_6$ 9I1 $9$ $18$ $C_3\times C_{18}$ 9I-1 3A $a^{4}c^{8}$
$C_2\times C_9^2:C_6$ 9I-1 $9$ $18$ $C_3\times C_{18}$ 9I1 3A $a^{2}c^{16}$
$C_2\times C_9^2:C_6$ 9J1 $9$ $18$ $C_3\times C_{18}$ 9J-1 3C $a^{4}b^{5}c^{8}$
$C_2\times C_9^2:C_6$ 9J-1 $9$ $18$ $C_3\times C_{18}$ 9J1 3C $a^{2}b^{7}c^{16}$
$C_2\times C_9^2:C_6$ 9K1 $9$ $18$ $C_3\times C_{18}$ 9K-1 3D $a^{4}bc^{8}$
$C_2\times C_9^2:C_6$ 9K-1 $9$ $18$ $C_3\times C_{18}$ 9K1 3D $a^{2}b^{5}c^{16}$
$C_2\times C_9^2:C_6$ 9L1 $9$ $18$ $C_3\times C_{18}$ 9L-1 3B $a^{4}b^{5}c^{6}$
$C_2\times C_9^2:C_6$ 9L-1 $9$ $18$ $C_3\times C_{18}$ 9L1 3B $a^{2}b^{7}c^{12}$
$C_2\times C_9^2:C_6$ 18A $18$ $6$ $C_9\times C_{18}$ 9A 6A $c$
$C_2\times C_9^2:C_6$ 18B $18$ $6$ $C_9\times C_{18}$ 9B 6A $b^{3}c$
$C_2\times C_9^2:C_6$ 18C $18$ $6$ $C_9\times C_{18}$ 9C 6A $b^{6}c$
$C_2\times C_9^2:C_6$ 18D $18$ $6$ $C_9\times C_{18}$ 9D 6B $bc^{3}$
$C_2\times C_9^2:C_6$ 18E $18$ $6$ $C_9\times C_{18}$ 9E 6B $b^{2}c^{3}$
$C_2\times C_9^2:C_6$ 18F $18$ $6$ $C_9\times C_{18}$ 9F 6B $bc^{9}$
$C_2\times C_9^2:C_6$ 18G1 $18$ $6$ $C_9\times C_{18}$ 9G1 6C $bc$
$C_2\times C_9^2:C_6$ 18G5 $18$ $6$ $C_9\times C_{18}$ 9G4 6C $bc^{7}$
$C_2\times C_9^2:C_6$ 18G7 $18$ $6$ $C_9\times C_{18}$ 9G2 6C $b^{4}c$
$C_2\times C_9^2:C_6$ 18H1 $18$ $6$ $C_9\times C_{18}$ 9H1 6D $b^{2}c$
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