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Elements of the group are displayed as words in the presentation $\langle a, b, c \mid a^{2}=b^{6}=c^{60}=[b,c]=1, b^{a}=b^{5}, c^{a}=c^{41} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 5P
$C_{60}:D_6$ 1A $1$ $1$ $C_{60}:D_6$ 1A 1A 1A $1$
$C_{60}:D_6$ 2A $2$ $1$ $C_{60}:D_6$ 1A 2A 2A $c^{30}$
$C_{60}:D_6$ 2B $2$ $1$ $C_{60}:D_6$ 1A 2B 2B $b^{3}$
$C_{60}:D_6$ 2C $2$ $1$ $C_{60}:D_6$ 1A 2C 2C $b^{3}c^{30}$
$C_{60}:D_6$ 2D $2$ $9$ $C_2^2\times C_{20}$ 1A 2D 2D $a$
$C_{60}:D_6$ 2E $2$ $9$ $C_2^2\times C_{20}$ 1A 2E 2E $ac^{50}$
$C_{60}:D_6$ 2F $2$ $9$ $C_2^2\times C_{20}$ 1A 2F 2F $ab$
$C_{60}:D_6$ 2G $2$ $9$ $C_2^2\times C_{20}$ 1A 2G 2G $abc^{50}$
$C_{60}:D_6$ 3A $3$ $2$ $C_6\times C_{60}$ 3A 1A 3A $b^{4}$
$C_{60}:D_6$ 3B $3$ $2$ $C_6\times C_{60}$ 3B 1A 3B $b^{4}c^{40}$
$C_{60}:D_6$ 3C $3$ $2$ $C_6\times C_{60}$ 3C 1A 3C $b^{4}c^{20}$
$C_{60}:D_6$ 3D $3$ $2$ $C_6\times C_{60}$ 3D 1A 3D $c^{40}$
$C_{60}:D_6$ 4A1 $4$ $1$ $C_{60}:D_6$ 2A 4A-1 4A1 $c^{15}$
$C_{60}:D_6$ 4A-1 $4$ $1$ $C_{60}:D_6$ 2A 4A1 4A-1 $c^{45}$
$C_{60}:D_6$ 4B1 $4$ $1$ $C_{60}:D_6$ 2A 4B-1 4B1 $b^{3}c^{15}$
$C_{60}:D_6$ 4B-1 $4$ $1$ $C_{60}:D_6$ 2A 4B1 4B-1 $b^{3}c^{45}$
$C_{60}:D_6$ 4C1 $4$ $9$ $C_2^2\times C_{20}$ 2A 4C-1 4C1 $ac^{25}$
$C_{60}:D_6$ 4C-1 $4$ $9$ $C_2^2\times C_{20}$ 2A 4C1 4C-1 $ac^{55}$
$C_{60}:D_6$ 4D1 $4$ $9$ $C_2^2\times C_{20}$ 2A 4D-1 4D1 $abc^{25}$
$C_{60}:D_6$ 4D-1 $4$ $9$ $C_2^2\times C_{20}$ 2A 4D1 4D-1 $abc^{55}$
$C_{60}:D_6$ 5A1 $5$ $1$ $C_{60}:D_6$ 5A2 5A-2 1A $c^{24}$
$C_{60}:D_6$ 5A-1 $5$ $1$ $C_{60}:D_6$ 5A-2 5A2 1A $c^{36}$
$C_{60}:D_6$ 5A2 $5$ $1$ $C_{60}:D_6$ 5A-1 5A1 1A $c^{48}$
$C_{60}:D_6$ 5A-2 $5$ $1$ $C_{60}:D_6$ 5A1 5A-1 1A $c^{12}$
$C_{60}:D_6$ 6A $6$ $2$ $C_6\times C_{60}$ 3A 2B 6A $b^{5}$
$C_{60}:D_6$ 6B $6$ $2$ $C_6\times C_{60}$ 3B 2B 6B $b^{5}c^{20}$
$C_{60}:D_6$ 6C $6$ $2$ $C_6\times C_{60}$ 3C 2C 6C $b^{5}c^{10}$
$C_{60}:D_6$ 6D $6$ $2$ $C_6\times C_{60}$ 3C 2B 6D $b^{5}c^{40}$
$C_{60}:D_6$ 6E $6$ $2$ $C_6\times C_{60}$ 3A 2C 6E $b^{5}c^{30}$
$C_{60}:D_6$ 6F $6$ $2$ $C_6\times C_{60}$ 3D 2B 6F $b^{3}c^{20}$
$C_{60}:D_6$ 6G $6$ $2$ $C_6\times C_{60}$ 3D 2C 6G $b^{3}c^{10}$
$C_{60}:D_6$ 6H $6$ $2$ $C_6\times C_{60}$ 3B 2C 6H $bc^{10}$
$C_{60}:D_6$ 6I $6$ $2$ $C_6\times C_{60}$ 3D 2A 6I $c^{10}$
$C_{60}:D_6$ 6J $6$ $2$ $C_6\times C_{60}$ 3C 2A 6J $b^{2}c^{10}$
$C_{60}:D_6$ 6K $6$ $2$ $C_6\times C_{60}$ 3B 2A 6K $b^{4}c^{10}$
$C_{60}:D_6$ 6L $6$ $2$ $C_6\times C_{60}$ 3A 2A 6L $b^{2}c^{30}$
$C_{60}:D_6$ 10A1 $10$ $1$ $C_{60}:D_6$ 5A1 10A3 2A $c^{42}$
$C_{60}:D_6$ 10A-1 $10$ $1$ $C_{60}:D_6$ 5A-1 10A-3 2A $c^{18}$
$C_{60}:D_6$ 10A3 $10$ $1$ $C_{60}:D_6$ 5A-2 10A-1 2A $c^{6}$
$C_{60}:D_6$ 10A-3 $10$ $1$ $C_{60}:D_6$ 5A2 10A1 2A $c^{54}$
$C_{60}:D_6$ 10B1 $10$ $1$ $C_{60}:D_6$ 5A-2 10B3 2B $b^{3}c^{36}$
$C_{60}:D_6$ 10B-1 $10$ $1$ $C_{60}:D_6$ 5A2 10B-3 2B $b^{3}c^{24}$
$C_{60}:D_6$ 10B3 $10$ $1$ $C_{60}:D_6$ 5A-1 10B-1 2B $b^{3}c^{48}$
$C_{60}:D_6$ 10B-3 $10$ $1$ $C_{60}:D_6$ 5A1 10B1 2B $b^{3}c^{12}$
$C_{60}:D_6$ 10C1 $10$ $1$ $C_{60}:D_6$ 5A-2 10C3 2C $b^{3}c^{6}$
$C_{60}:D_6$ 10C-1 $10$ $1$ $C_{60}:D_6$ 5A2 10C-3 2C $b^{3}c^{54}$
$C_{60}:D_6$ 10C3 $10$ $1$ $C_{60}:D_6$ 5A-1 10C-1 2C $b^{3}c^{18}$
$C_{60}:D_6$ 10C-3 $10$ $1$ $C_{60}:D_6$ 5A1 10C1 2C $b^{3}c^{42}$
$C_{60}:D_6$ 10D1 $10$ $9$ $C_2^2\times C_{20}$ 5A1 10D3 2D $ac^{12}$
$C_{60}:D_6$ 10D-1 $10$ $9$ $C_2^2\times C_{20}$ 5A-1 10D-3 2D $ac^{8}$
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