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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{60}=[a,b]=[a,c]=[b,c]=[c,d]=1, d^{a}=d^{41}, d^{b}=d^{31} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 5P
$C_{60}:C_2^3$ 1A $1$ $1$ $C_{60}:C_2^3$ 1A 1A 1A $1$
$C_{60}:C_2^3$ 2A $2$ $1$ $C_{60}:C_2^3$ 1A 2A 2A $d^{30}$
$C_{60}:C_2^3$ 2B $2$ $1$ $C_{60}:C_2^3$ 1A 2B 2B $c$
$C_{60}:C_2^3$ 2C $2$ $1$ $C_{60}:C_2^3$ 1A 2C 2C $cd^{30}$
$C_{60}:C_2^3$ 2D $2$ $2$ $C_{15}:C_2^4$ 1A 2D 2D $b$
$C_{60}:C_2^3$ 2E $2$ $2$ $C_{15}:C_2^4$ 1A 2E 2E $bd^{45}$
$C_{60}:C_2^3$ 2F $2$ $2$ $C_{15}:C_2^4$ 1A 2F 2F $bc$
$C_{60}:C_2^3$ 2G $2$ $2$ $C_{15}:C_2^4$ 1A 2G 2G $bcd^{45}$
$C_{60}:C_2^3$ 2H $2$ $3$ $C_{20}:C_2^3$ 1A 2H 2H $a$
$C_{60}:C_2^3$ 2I $2$ $3$ $C_{20}:C_2^3$ 1A 2I 2I $ad^{50}$
$C_{60}:C_2^3$ 2J $2$ $3$ $C_{20}:C_2^3$ 1A 2J 2J $ac$
$C_{60}:C_2^3$ 2K $2$ $3$ $C_{20}:C_2^3$ 1A 2K 2K $acd^{50}$
$C_{60}:C_2^3$ 2L $2$ $6$ $C_2^3\times C_{10}$ 1A 2L 2L $ab$
$C_{60}:C_2^3$ 2M $2$ $6$ $C_2^3\times C_{10}$ 1A 2M 2M $abd^{25}$
$C_{60}:C_2^3$ 2N $2$ $6$ $C_2^3\times C_{10}$ 1A 2N 2N $abc$
$C_{60}:C_2^3$ 2O $2$ $6$ $C_2^3\times C_{10}$ 1A 2O 2O $abcd^{25}$
$C_{60}:C_2^3$ 3A $3$ $2$ $D_4\times C_{30}$ 3A 1A 3A $d^{40}$
$C_{60}:C_2^3$ 4A $4$ $2$ $D_6\times C_{20}$ 2A 4A 4A $d^{15}$
$C_{60}:C_2^3$ 4B $4$ $2$ $D_6\times C_{20}$ 2A 4B 4B $cd^{15}$
$C_{60}:C_2^3$ 4C $4$ $6$ $C_2^2\times C_{20}$ 2A 4C 4C $ad^{25}$
$C_{60}:C_2^3$ 4D $4$ $6$ $C_2^2\times C_{20}$ 2A 4D 4D $acd^{25}$
$C_{60}:C_2^3$ 5A1 $5$ $1$ $C_{60}:C_2^3$ 5A2 5A-2 1A $d^{24}$
$C_{60}:C_2^3$ 5A-1 $5$ $1$ $C_{60}:C_2^3$ 5A-2 5A2 1A $d^{36}$
$C_{60}:C_2^3$ 5A2 $5$ $1$ $C_{60}:C_2^3$ 5A-1 5A1 1A $d^{48}$
$C_{60}:C_2^3$ 5A-2 $5$ $1$ $C_{60}:C_2^3$ 5A1 5A-1 1A $d^{12}$
$C_{60}:C_2^3$ 6A $6$ $2$ $D_4\times C_{30}$ 3A 2B 6A $cd^{20}$
$C_{60}:C_2^3$ 6B $6$ $2$ $D_4\times C_{30}$ 3A 2C 6B $cd^{10}$
$C_{60}:C_2^3$ 6C $6$ $2$ $D_4\times C_{30}$ 3A 2A 6C $d^{10}$
$C_{60}:C_2^3$ 6D $6$ $4$ $C_2^2\times C_{30}$ 3A 2D 6D $bd^{20}$
$C_{60}:C_2^3$ 6E $6$ $4$ $C_2^2\times C_{30}$ 3A 2E 6E $bd^{5}$
$C_{60}:C_2^3$ 6F $6$ $4$ $C_2^2\times C_{30}$ 3A 2F 6F $bcd^{20}$
$C_{60}:C_2^3$ 6G $6$ $4$ $C_2^2\times C_{30}$ 3A 2G 6G $bcd^{5}$
$C_{60}:C_2^3$ 10A1 $10$ $1$ $C_{60}:C_2^3$ 5A1 10A3 2A $d^{42}$
$C_{60}:C_2^3$ 10A-1 $10$ $1$ $C_{60}:C_2^3$ 5A-1 10A-3 2A $d^{18}$
$C_{60}:C_2^3$ 10A3 $10$ $1$ $C_{60}:C_2^3$ 5A-2 10A-1 2A $d^{6}$
$C_{60}:C_2^3$ 10A-3 $10$ $1$ $C_{60}:C_2^3$ 5A2 10A1 2A $d^{54}$
$C_{60}:C_2^3$ 10B1 $10$ $1$ $C_{60}:C_2^3$ 5A1 10B3 2B $cd^{12}$
$C_{60}:C_2^3$ 10B-1 $10$ $1$ $C_{60}:C_2^3$ 5A-1 10B-3 2B $cd^{48}$
$C_{60}:C_2^3$ 10B3 $10$ $1$ $C_{60}:C_2^3$ 5A-2 10B-1 2B $cd^{36}$
$C_{60}:C_2^3$ 10B-3 $10$ $1$ $C_{60}:C_2^3$ 5A2 10B1 2B $cd^{24}$
$C_{60}:C_2^3$ 10C1 $10$ $1$ $C_{60}:C_2^3$ 5A-2 10C3 2C $cd^{6}$
$C_{60}:C_2^3$ 10C-1 $10$ $1$ $C_{60}:C_2^3$ 5A2 10C-3 2C $cd^{54}$
$C_{60}:C_2^3$ 10C3 $10$ $1$ $C_{60}:C_2^3$ 5A-1 10C-1 2C $cd^{18}$
$C_{60}:C_2^3$ 10C-3 $10$ $1$ $C_{60}:C_2^3$ 5A1 10C1 2C $cd^{42}$
$C_{60}:C_2^3$ 10D1 $10$ $2$ $C_{15}:C_2^4$ 5A1 10D3 2D $bd^{12}$
$C_{60}:C_2^3$ 10D-1 $10$ $2$ $C_{15}:C_2^4$ 5A-1 10D-3 2D $bd^{48}$
$C_{60}:C_2^3$ 10D3 $10$ $2$ $C_{15}:C_2^4$ 5A-2 10D-1 2D $bd^{36}$
$C_{60}:C_2^3$ 10D-3 $10$ $2$ $C_{15}:C_2^4$ 5A2 10D1 2D $bd^{24}$
$C_{60}:C_2^3$ 10E1 $10$ $2$ $C_{15}:C_2^4$ 5A-1 10E3 2E $bd^{33}$
$C_{60}:C_2^3$ 10E-1 $10$ $2$ $C_{15}:C_2^4$ 5A1 10E-3 2E $bd^{57}$
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