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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{2}=b^{3}=c^{3}=d^{60}=[a,c]=[b,c]=[c,d]=1, b^{a}=b^{2}, d^{a}=d^{41}, d^{b}=c^{2}d \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 5P
$C_3^2:S_3\times C_{20}$ 1A $1$ $1$ $C_3^2:S_3\times C_{20}$ 1A 1A 1A $1$
$C_3^2:S_3\times C_{20}$ 2A $2$ $1$ $C_3^2:S_3\times C_{20}$ 1A 2A 2A $d^{30}$
$C_3^2:S_3\times C_{20}$ 2B $2$ $9$ $C_2\times C_{60}$ 1A 2B 2B $a$
$C_3^2:S_3\times C_{20}$ 2C $2$ $9$ $C_2\times C_{60}$ 1A 2C 2C $ad^{50}$
$C_3^2:S_3\times C_{20}$ 3A1 $3$ $1$ $C_3^2:S_3\times C_{20}$ 3A-1 1A 3A-1 $c$
$C_3^2:S_3\times C_{20}$ 3A-1 $3$ $1$ $C_3^2:S_3\times C_{20}$ 3A1 1A 3A1 $c^{2}$
$C_3^2:S_3\times C_{20}$ 3B $3$ $6$ $C_3\times C_{60}$ 3B 1A 3B $d^{20}$
$C_3^2:S_3\times C_{20}$ 3C $3$ $6$ $C_3\times C_{60}$ 3C 1A 3C $b^{2}c^{2}d^{20}$
$C_3^2:S_3\times C_{20}$ 3D $3$ $6$ $C_3\times C_{60}$ 3D 1A 3D $bcd^{20}$
$C_3^2:S_3\times C_{20}$ 3E $3$ $6$ $C_3\times C_{60}$ 3E 1A 3E $b^{2}$
$C_3^2:S_3\times C_{20}$ 4A1 $4$ $1$ $C_3^2:S_3\times C_{20}$ 2A 4A-1 4A1 $d^{45}$
$C_3^2:S_3\times C_{20}$ 4A-1 $4$ $1$ $C_3^2:S_3\times C_{20}$ 2A 4A1 4A-1 $d^{15}$
$C_3^2:S_3\times C_{20}$ 4B1 $4$ $9$ $C_2\times C_{60}$ 2A 4B-1 4B1 $ad^{55}$
$C_3^2:S_3\times C_{20}$ 4B-1 $4$ $9$ $C_2\times C_{60}$ 2A 4B1 4B-1 $ad^{25}$
$C_3^2:S_3\times C_{20}$ 5A1 $5$ $1$ $C_3^2:S_3\times C_{20}$ 5A2 5A-2 1A $d^{12}$
$C_3^2:S_3\times C_{20}$ 5A-1 $5$ $1$ $C_3^2:S_3\times C_{20}$ 5A-2 5A2 1A $d^{48}$
$C_3^2:S_3\times C_{20}$ 5A2 $5$ $1$ $C_3^2:S_3\times C_{20}$ 5A-1 5A1 1A $d^{24}$
$C_3^2:S_3\times C_{20}$ 5A-2 $5$ $1$ $C_3^2:S_3\times C_{20}$ 5A1 5A-1 1A $d^{36}$
$C_3^2:S_3\times C_{20}$ 6A1 $6$ $1$ $C_3^2:S_3\times C_{20}$ 3A-1 2A 6A-1 $cd^{30}$
$C_3^2:S_3\times C_{20}$ 6A-1 $6$ $1$ $C_3^2:S_3\times C_{20}$ 3A1 2A 6A1 $c^{2}d^{30}$
$C_3^2:S_3\times C_{20}$ 6B $6$ $6$ $C_3\times C_{60}$ 3B 2A 6B $d^{10}$
$C_3^2:S_3\times C_{20}$ 6C $6$ $6$ $C_3\times C_{60}$ 3C 2A 6C $bd^{10}$
$C_3^2:S_3\times C_{20}$ 6D $6$ $6$ $C_3\times C_{60}$ 3D 2A 6D $b^{2}d^{10}$
$C_3^2:S_3\times C_{20}$ 6E $6$ $6$ $C_3\times C_{60}$ 3E 2A 6E $bd^{30}$
$C_3^2:S_3\times C_{20}$ 6F1 $6$ $9$ $C_2\times C_{60}$ 3A1 2B 6F-1 $ac^{2}$
$C_3^2:S_3\times C_{20}$ 6F-1 $6$ $9$ $C_2\times C_{60}$ 3A-1 2B 6F1 $ac$
$C_3^2:S_3\times C_{20}$ 6G1 $6$ $9$ $C_2\times C_{60}$ 3A1 2C 6G-1 $ac^{2}d^{50}$
$C_3^2:S_3\times C_{20}$ 6G-1 $6$ $9$ $C_2\times C_{60}$ 3A-1 2C 6G1 $acd^{50}$
$C_3^2:S_3\times C_{20}$ 10A1 $10$ $1$ $C_3^2:S_3\times C_{20}$ 5A-2 10A3 2A $d^{18}$
$C_3^2:S_3\times C_{20}$ 10A-1 $10$ $1$ $C_3^2:S_3\times C_{20}$ 5A2 10A-3 2A $d^{42}$
$C_3^2:S_3\times C_{20}$ 10A3 $10$ $1$ $C_3^2:S_3\times C_{20}$ 5A-1 10A-1 2A $d^{54}$
$C_3^2:S_3\times C_{20}$ 10A-3 $10$ $1$ $C_3^2:S_3\times C_{20}$ 5A1 10A1 2A $d^{6}$
$C_3^2:S_3\times C_{20}$ 10B1 $10$ $9$ $C_2\times C_{60}$ 5A1 10B3 2B $ad^{36}$
$C_3^2:S_3\times C_{20}$ 10B-1 $10$ $9$ $C_2\times C_{60}$ 5A-1 10B-3 2B $ad^{24}$
$C_3^2:S_3\times C_{20}$ 10B3 $10$ $9$ $C_2\times C_{60}$ 5A-2 10B-1 2B $ad^{48}$
$C_3^2:S_3\times C_{20}$ 10B-3 $10$ $9$ $C_2\times C_{60}$ 5A2 10B1 2B $ad^{12}$
$C_3^2:S_3\times C_{20}$ 10C1 $10$ $9$ $C_2\times C_{60}$ 5A1 10C3 2C $ad^{26}$
$C_3^2:S_3\times C_{20}$ 10C-1 $10$ $9$ $C_2\times C_{60}$ 5A-1 10C-3 2C $ad^{14}$
$C_3^2:S_3\times C_{20}$ 10C3 $10$ $9$ $C_2\times C_{60}$ 5A-2 10C-1 2C $ad^{38}$
$C_3^2:S_3\times C_{20}$ 10C-3 $10$ $9$ $C_2\times C_{60}$ 5A2 10C1 2C $ad^{2}$
$C_3^2:S_3\times C_{20}$ 12A1 $12$ $1$ $C_3^2:S_3\times C_{20}$ 6A1 4A1 12A5 $c^{2}d^{15}$
$C_3^2:S_3\times C_{20}$ 12A-1 $12$ $1$ $C_3^2:S_3\times C_{20}$ 6A-1 4A-1 12A-5 $cd^{45}$
$C_3^2:S_3\times C_{20}$ 12A5 $12$ $1$ $C_3^2:S_3\times C_{20}$ 6A-1 4A1 12A1 $cd^{15}$
$C_3^2:S_3\times C_{20}$ 12A-5 $12$ $1$ $C_3^2:S_3\times C_{20}$ 6A1 4A-1 12A-1 $c^{2}d^{45}$
$C_3^2:S_3\times C_{20}$ 12B1 $12$ $6$ $C_3\times C_{60}$ 6B 4A-1 12B1 $d^{5}$
$C_3^2:S_3\times C_{20}$ 12B-1 $12$ $6$ $C_3\times C_{60}$ 6B 4A1 12B-1 $d^{55}$
$C_3^2:S_3\times C_{20}$ 12C1 $12$ $6$ $C_3\times C_{60}$ 6C 4A-1 12C1 $b^{2}c^{2}d^{5}$
$C_3^2:S_3\times C_{20}$ 12C-1 $12$ $6$ $C_3\times C_{60}$ 6C 4A1 12C-1 $bd^{55}$
$C_3^2:S_3\times C_{20}$ 12D1 $12$ $6$ $C_3\times C_{60}$ 6D 4A-1 12D1 $bcd^{5}$
$C_3^2:S_3\times C_{20}$ 12D-1 $12$ $6$ $C_3\times C_{60}$ 6D 4A1 12D-1 $b^{2}d^{55}$
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