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Elements of the group are displayed as words in the presentation $\langle a, b, c \mid c^{12}=1, a^{6}=c^{6}, b^{6}=c^{6}, b^{a}=b^{5}, c^{a}=c^{7}, c^{b}=c^{11} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_{12}.S_3^2$ 1A $1$ $1$ $C_{12}.S_3^2$ 1A 1A $1$
$C_{12}.S_3^2$ 2A $2$ $1$ $C_{12}.S_3^2$ 1A 2A $c^{6}$
$C_{12}.S_3^2$ 2B $2$ $9$ $C_6\times Q_8$ 1A 2B $a^{3}bc^{6}$
$C_{12}.S_3^2$ 2C $2$ $9$ $C_6\times Q_8$ 1A 2C $a^{3}bc^{8}$
$C_{12}.S_3^2$ 3A1 $3$ $1$ $C_{12}.S_3^2$ 3A-1 1A $a^{2}c^{6}$
$C_{12}.S_3^2$ 3A-1 $3$ $1$ $C_{12}.S_3^2$ 3A1 1A $a^{4}$
$C_{12}.S_3^2$ 3B $3$ $2$ $C_3^3:Q_8$ 3B 1A $c^{4}$
$C_{12}.S_3^2$ 3C $3$ $2$ $C_3^3:Q_8$ 3C 1A $b^{2}c^{6}$
$C_{12}.S_3^2$ 3D1 $3$ $2$ $C_3^3:Q_8$ 3D-1 1A $a^{2}c^{10}$
$C_{12}.S_3^2$ 3D-1 $3$ $2$ $C_3^3:Q_8$ 3D1 1A $a^{4}c^{8}$
$C_{12}.S_3^2$ 3E1 $3$ $2$ $C_3^3:Q_8$ 3E-1 1A $a^{2}b^{2}$
$C_{12}.S_3^2$ 3E-1 $3$ $2$ $C_3^3:Q_8$ 3E1 1A $a^{4}b^{4}$
$C_{12}.S_3^2$ 3F $3$ $4$ $C_3^2\times C_{12}$ 3F 1A $b^{2}c^{10}$
$C_{12}.S_3^2$ 3G1 $3$ $4$ $C_3^2\times C_{12}$ 3G-1 1A $a^{2}b^{2}c^{4}$
$C_{12}.S_3^2$ 3G-1 $3$ $4$ $C_3^2\times C_{12}$ 3G1 1A $a^{4}b^{4}c^{8}$
$C_{12}.S_3^2$ 4A $4$ $2$ $C_3:S_3\times C_{12}$ 2A 4A $c^{3}$
$C_{12}.S_3^2$ 4B $4$ $6$ $S_3\times C_{12}$ 2A 4B $a^{3}$
$C_{12}.S_3^2$ 4C $4$ $6$ $S_3\times C_{12}$ 2A 4C $b^{3}$
$C_{12}.S_3^2$ 4D $4$ $6$ $S_3\times C_{12}$ 2A 4D $a^{3}c^{3}$
$C_{12}.S_3^2$ 4E $4$ $6$ $S_3\times C_{12}$ 2A 4E $b^{3}c$
$C_{12}.S_3^2$ 4F $4$ $18$ $C_2\times C_{12}$ 2A 4F $a^{3}bc$
$C_{12}.S_3^2$ 6A1 $6$ $1$ $C_{12}.S_3^2$ 3A1 2A $a^{4}c^{6}$
$C_{12}.S_3^2$ 6A-1 $6$ $1$ $C_{12}.S_3^2$ 3A-1 2A $a^{2}$
$C_{12}.S_3^2$ 6B $6$ $2$ $C_3^3:Q_8$ 3B 2A $c^{2}$
$C_{12}.S_3^2$ 6C $6$ $2$ $C_3^3:Q_8$ 3C 2A $b^{4}c^{6}$
$C_{12}.S_3^2$ 6D1 $6$ $2$ $C_3^3:Q_8$ 3D1 2A $a^{4}c^{2}$
$C_{12}.S_3^2$ 6D-1 $6$ $2$ $C_3^3:Q_8$ 3D-1 2A $a^{2}c^{4}$
$C_{12}.S_3^2$ 6E1 $6$ $2$ $C_3^3:Q_8$ 3E1 2A $a^{4}b^{4}c^{6}$
$C_{12}.S_3^2$ 6E-1 $6$ $2$ $C_3^3:Q_8$ 3E-1 2A $a^{2}b^{2}c^{6}$
$C_{12}.S_3^2$ 6F $6$ $4$ $C_3^2\times C_{12}$ 3F 2A $b^{4}c^{2}$
$C_{12}.S_3^2$ 6G1 $6$ $4$ $C_3^2\times C_{12}$ 3G1 2A $a^{4}b^{4}c^{2}$
$C_{12}.S_3^2$ 6G-1 $6$ $4$ $C_3^2\times C_{12}$ 3G-1 2A $a^{2}b^{2}c^{10}$
$C_{12}.S_3^2$ 6H1 $6$ $9$ $C_6\times Q_8$ 3A1 2B $ab$
$C_{12}.S_3^2$ 6H-1 $6$ $9$ $C_6\times Q_8$ 3A-1 2B $a^{5}b$
$C_{12}.S_3^2$ 6I1 $6$ $9$ $C_6\times Q_8$ 3A1 2C $abc^{2}$
$C_{12}.S_3^2$ 6I-1 $6$ $9$ $C_6\times Q_8$ 3A-1 2C $a^{5}bc^{2}$
$C_{12}.S_3^2$ 12A1 $12$ $2$ $C_3:S_3\times C_{12}$ 6A1 4A $a^{2}c^{3}$
$C_{12}.S_3^2$ 12A-1 $12$ $2$ $C_3:S_3\times C_{12}$ 6A-1 4A $a^{4}c^{3}$
$C_{12}.S_3^2$ 12B $12$ $4$ $C_3^2\times C_{12}$ 6B 4A $c$
$C_{12}.S_3^2$ 12C $12$ $4$ $C_3^2\times C_{12}$ 6C 4A $b^{2}c^{3}$
$C_{12}.S_3^2$ 12D1 $12$ $4$ $C_3^2\times C_{12}$ 6F 4A $b^{2}c$
$C_{12}.S_3^2$ 12D-1 $12$ $4$ $C_3^2\times C_{12}$ 6F 4A $b^{2}c^{5}$
$C_{12}.S_3^2$ 12E1 $12$ $4$ $C_3^2\times C_{12}$ 6D1 4A $a^{2}c$
$C_{12}.S_3^2$ 12E-1 $12$ $4$ $C_3^2\times C_{12}$ 6D-1 4A $a^{4}c$
$C_{12}.S_3^2$ 12F1 $12$ $4$ $C_3^2\times C_{12}$ 6E1 4A $a^{2}b^{2}c^{3}$
$C_{12}.S_3^2$ 12F-1 $12$ $4$ $C_3^2\times C_{12}$ 6E-1 4A $a^{4}b^{2}c^{3}$
$C_{12}.S_3^2$ 12G1 $12$ $4$ $C_3^2\times C_{12}$ 6G1 4A $a^{2}b^{2}c$
$C_{12}.S_3^2$ 12G-1 $12$ $4$ $C_3^2\times C_{12}$ 6G-1 4A $a^{4}b^{2}c$
$C_{12}.S_3^2$ 12G5 $12$ $4$ $C_3^2\times C_{12}$ 6G-1 4A $a^{4}b^{2}c^{5}$
$C_{12}.S_3^2$ 12G-5 $12$ $4$ $C_3^2\times C_{12}$ 6G1 4A $a^{2}b^{2}c^{5}$
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