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Elements of the group are displayed as words in the presentation $\langle a, b, c \mid a^{2}=b^{12}=c^{60}=[b,c]=1, b^{a}=b^{11}c^{15}, c^{a}=c^{41} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 5P
$C_{12}^2:C_{10}$ 1A $1$ $1$ $C_{12}^2:C_{10}$ 1A 1A 1A $1$
$C_{12}^2:C_{10}$ 2A $2$ $1$ $C_{12}^2:C_{10}$ 1A 2A 2A $c^{30}$
$C_{12}^2:C_{10}$ 2B $2$ $2$ $C_{12}\times C_{60}$ 1A 2B 2B $b^{6}$
$C_{12}^2:C_{10}$ 2C $2$ $36$ $C_2\times C_{20}$ 1A 2C 2C $a$
$C_{12}^2:C_{10}$ 3A $3$ $2$ $C_{12}\times C_{60}$ 3A 1A 3A $c^{20}$
$C_{12}^2:C_{10}$ 3B $3$ $2$ $C_{12}\times C_{60}$ 3B 1A 3B $b^{8}c^{20}$
$C_{12}^2:C_{10}$ 3C $3$ $2$ $C_{12}\times C_{60}$ 3C 1A 3C $b^{4}c^{20}$
$C_{12}^2:C_{10}$ 3D $3$ $2$ $C_{12}\times C_{60}$ 3D 1A 3D $b^{8}$
$C_{12}^2:C_{10}$ 4A1 $4$ $1$ $C_{12}^2:C_{10}$ 2A 4A-1 4A1 $c^{15}$
$C_{12}^2:C_{10}$ 4A-1 $4$ $1$ $C_{12}^2:C_{10}$ 2A 4A1 4A-1 $c^{45}$
$C_{12}^2:C_{10}$ 4B $4$ $2$ $C_{12}\times C_{60}$ 2A 4B 4B $b^{6}c^{15}$
$C_{12}^2:C_{10}$ 4C1 $4$ $2$ $C_{12}\times C_{60}$ 2B 4C-1 4C1 $b^{3}$
$C_{12}^2:C_{10}$ 4C-1 $4$ $2$ $C_{12}\times C_{60}$ 2B 4C1 4C-1 $b^{9}$
$C_{12}^2:C_{10}$ 4D1 $4$ $2$ $C_{12}\times C_{60}$ 2B 4D-1 4D1 $b^{3}c^{30}$
$C_{12}^2:C_{10}$ 4D-1 $4$ $2$ $C_{12}\times C_{60}$ 2B 4D1 4D-1 $b^{9}c^{30}$
$C_{12}^2:C_{10}$ 4E $4$ $36$ $C_2\times C_{20}$ 2A 4E 4E $ac^{25}$
$C_{12}^2:C_{10}$ 5A1 $5$ $1$ $C_{12}^2:C_{10}$ 5A2 5A-2 1A $c^{24}$
$C_{12}^2:C_{10}$ 5A-1 $5$ $1$ $C_{12}^2:C_{10}$ 5A-2 5A2 1A $c^{36}$
$C_{12}^2:C_{10}$ 5A2 $5$ $1$ $C_{12}^2:C_{10}$ 5A-1 5A1 1A $c^{48}$
$C_{12}^2:C_{10}$ 5A-2 $5$ $1$ $C_{12}^2:C_{10}$ 5A1 5A-1 1A $c^{12}$
$C_{12}^2:C_{10}$ 6A $6$ $2$ $C_{12}\times C_{60}$ 3A 2A 6A $c^{10}$
$C_{12}^2:C_{10}$ 6B $6$ $2$ $C_{12}\times C_{60}$ 3B 2A 6B $b^{4}c^{10}$
$C_{12}^2:C_{10}$ 6C $6$ $2$ $C_{12}\times C_{60}$ 3C 2A 6C $b^{8}c^{10}$
$C_{12}^2:C_{10}$ 6D $6$ $2$ $C_{12}\times C_{60}$ 3D 2A 6D $b^{4}c^{30}$
$C_{12}^2:C_{10}$ 6E1 $6$ $2$ $C_{12}\times C_{60}$ 3D 2B 6E-1 $b^{10}$
$C_{12}^2:C_{10}$ 6E-1 $6$ $2$ $C_{12}\times C_{60}$ 3D 2B 6E1 $b^{2}$
$C_{12}^2:C_{10}$ 6F1 $6$ $2$ $C_{12}\times C_{60}$ 3B 2B 6F-1 $b^{10}c^{40}$
$C_{12}^2:C_{10}$ 6F-1 $6$ $2$ $C_{12}\times C_{60}$ 3B 2B 6F1 $b^{2}c^{20}$
$C_{12}^2:C_{10}$ 6G1 $6$ $2$ $C_{12}\times C_{60}$ 3C 2B 6G-1 $b^{10}c^{20}$
$C_{12}^2:C_{10}$ 6G-1 $6$ $2$ $C_{12}\times C_{60}$ 3C 2B 6G1 $b^{2}c^{40}$
$C_{12}^2:C_{10}$ 6H1 $6$ $2$ $C_{12}\times C_{60}$ 3A 2B 6H-1 $b^{6}c^{40}$
$C_{12}^2:C_{10}$ 6H-1 $6$ $2$ $C_{12}\times C_{60}$ 3A 2B 6H1 $b^{6}c^{20}$
$C_{12}^2:C_{10}$ 8A1 $8$ $36$ $C_{40}$ 4A1 8A-1 8A1 $abc^{30}$
$C_{12}^2:C_{10}$ 8A-1 $8$ $36$ $C_{40}$ 4A-1 8A1 8A-1 $abc^{15}$
$C_{12}^2:C_{10}$ 10A1 $10$ $1$ $C_{12}^2:C_{10}$ 5A1 10A3 2A $c^{42}$
$C_{12}^2:C_{10}$ 10A-1 $10$ $1$ $C_{12}^2:C_{10}$ 5A-1 10A-3 2A $c^{18}$
$C_{12}^2:C_{10}$ 10A3 $10$ $1$ $C_{12}^2:C_{10}$ 5A-2 10A-1 2A $c^{6}$
$C_{12}^2:C_{10}$ 10A-3 $10$ $1$ $C_{12}^2:C_{10}$ 5A2 10A1 2A $c^{54}$
$C_{12}^2:C_{10}$ 10B1 $10$ $2$ $C_{12}\times C_{60}$ 5A1 10B3 2B $b^{6}c^{12}$
$C_{12}^2:C_{10}$ 10B-1 $10$ $2$ $C_{12}\times C_{60}$ 5A-1 10B-3 2B $b^{6}c^{48}$
$C_{12}^2:C_{10}$ 10B3 $10$ $2$ $C_{12}\times C_{60}$ 5A-2 10B-1 2B $b^{6}c^{36}$
$C_{12}^2:C_{10}$ 10B-3 $10$ $2$ $C_{12}\times C_{60}$ 5A2 10B1 2B $b^{6}c^{24}$
$C_{12}^2:C_{10}$ 10C1 $10$ $36$ $C_2\times C_{20}$ 5A1 10C3 2C $ac^{12}$
$C_{12}^2:C_{10}$ 10C-1 $10$ $36$ $C_2\times C_{20}$ 5A-1 10C-3 2C $ac^{8}$
$C_{12}^2:C_{10}$ 10C3 $10$ $36$ $C_2\times C_{20}$ 5A-2 10C-1 2C $ac^{36}$
$C_{12}^2:C_{10}$ 10C-3 $10$ $36$ $C_2\times C_{20}$ 5A2 10C1 2C $ac^{4}$
$C_{12}^2:C_{10}$ 12A1 $12$ $2$ $C_{12}\times C_{60}$ 6A 4A1 12A1 $c^{5}$
$C_{12}^2:C_{10}$ 12A-1 $12$ $2$ $C_{12}\times C_{60}$ 6A 4A-1 12A-1 $c^{55}$
$C_{12}^2:C_{10}$ 12B1 $12$ $2$ $C_{12}\times C_{60}$ 6B 4A1 12B1 $b^{8}c^{5}$
$C_{12}^2:C_{10}$ 12B-1 $12$ $2$ $C_{12}\times C_{60}$ 6B 4A-1 12B-1 $b^{4}c^{55}$
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