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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{12}=b^{5}=c^{15}=d^{15}=[b,c]=[b,d]=[c,d]=1, b^{a}=c^{6}d^{6}, c^{a}=b^{4}c^{13}d^{6}, d^{a}=b^{3}c^{4}d^{7} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 5P
$(C_5\times C_{15}^2):C_{12}$ 1A $1$ $1$ $(C_5\times C_{15}^2):C_{12}$ 1A 1A 1A $1$
$(C_5\times C_{15}^2):C_{12}$ 2A $2$ $125$ $C_4\times \He_3$ 1A 2A 2A $a^{6}$
$(C_5\times C_{15}^2):C_{12}$ 3A1 $3$ $1$ $(C_5\times C_{15}^2):C_{12}$ 3A-1 1A 3A-1 $c^{10}$
$(C_5\times C_{15}^2):C_{12}$ 3A-1 $3$ $1$ $(C_5\times C_{15}^2):C_{12}$ 3A1 1A 3A1 $c^{5}$
$(C_5\times C_{15}^2):C_{12}$ 3B1 $3$ $3$ $C_3^2\times C_5^2:D_5.C_2$ 3B-1 1A 3B-1 $d^{5}$
$(C_5\times C_{15}^2):C_{12}$ 3B-1 $3$ $3$ $C_3^2\times C_5^2:D_5.C_2$ 3B1 1A 3B1 $d^{10}$
$(C_5\times C_{15}^2):C_{12}$ 3C1 $3$ $75$ $C_3^2\times F_5$ 3C-1 1A 3C-1 $a^{8}b^{4}c^{13}d^{5}$
$(C_5\times C_{15}^2):C_{12}$ 3C-1 $3$ $75$ $C_3^2\times F_5$ 3C1 1A 3C1 $a^{4}bc^{9}d$
$(C_5\times C_{15}^2):C_{12}$ 3D1 $3$ $75$ $C_3^2\times F_5$ 3D-1 1A 3D-1 $a^{8}bc^{13}d^{13}$
$(C_5\times C_{15}^2):C_{12}$ 3D-1 $3$ $75$ $C_3^2\times F_5$ 3D1 1A 3D1 $a^{4}b^{3}cd^{5}$
$(C_5\times C_{15}^2):C_{12}$ 3E1 $3$ $75$ $C_3^2\times F_5$ 3E-1 1A 3E-1 $a^{8}bc^{10}d^{9}$
$(C_5\times C_{15}^2):C_{12}$ 3E-1 $3$ $75$ $C_3^2\times F_5$ 3E1 1A 3E1 $a^{4}bc^{11}d^{12}$
$(C_5\times C_{15}^2):C_{12}$ 4A1 $4$ $125$ $C_4\times \He_3$ 2A 4A-1 4A1 $a^{9}$
$(C_5\times C_{15}^2):C_{12}$ 4A-1 $4$ $125$ $C_4\times \He_3$ 2A 4A1 4A-1 $a^{3}$
$(C_5\times C_{15}^2):C_{12}$ 5A $5$ $4$ $C_{15}^2:C_{15}$ 5A 5A 1A $b^{3}d^{3}$
$(C_5\times C_{15}^2):C_{12}$ 5B $5$ $12$ $C_5\times C_{15}^2$ 5B 5B 1A $d^{3}$
$(C_5\times C_{15}^2):C_{12}$ 5C $5$ $12$ $C_5\times C_{15}^2$ 5C 5C 1A $c^{3}$
$(C_5\times C_{15}^2):C_{12}$ 5D $5$ $12$ $C_5\times C_{15}^2$ 5D 5D 1A $c^{3}d^{9}$
$(C_5\times C_{15}^2):C_{12}$ 5E $5$ $12$ $C_5\times C_{15}^2$ 5E 5E 1A $c^{3}d^{3}$
$(C_5\times C_{15}^2):C_{12}$ 5F $5$ $12$ $C_5\times C_{15}^2$ 5F 5F 1A $b^{3}c^{3}$
$(C_5\times C_{15}^2):C_{12}$ 5G $5$ $12$ $C_5\times C_{15}^2$ 5G 5G 1A $c^{9}d^{6}$
$(C_5\times C_{15}^2):C_{12}$ 5H $5$ $12$ $C_5\times C_{15}^2$ 5H 5H 1A $b^{3}d^{6}$
$(C_5\times C_{15}^2):C_{12}$ 5I $5$ $12$ $C_5\times C_{15}^2$ 5I 5I 1A $b^{3}c^{6}$
$(C_5\times C_{15}^2):C_{12}$ 5J $5$ $12$ $C_5\times C_{15}^2$ 5J 5J 1A $bd^{3}$
$(C_5\times C_{15}^2):C_{12}$ 5K $5$ $12$ $C_5\times C_{15}^2$ 5K 5K 1A $b^{3}c^{3}d^{3}$
$(C_5\times C_{15}^2):C_{12}$ 6A1 $6$ $125$ $C_4\times \He_3$ 3A1 2A 6A-1 $a^{6}c^{5}$
$(C_5\times C_{15}^2):C_{12}$ 6A-1 $6$ $125$ $C_4\times \He_3$ 3A-1 2A 6A1 $a^{6}c^{10}$
$(C_5\times C_{15}^2):C_{12}$ 6B1 $6$ $375$ $C_3\times C_{12}$ 3C-1 2A 6B-1 $a^{2}bc^{4}d^{8}$
$(C_5\times C_{15}^2):C_{12}$ 6B-1 $6$ $375$ $C_3\times C_{12}$ 3C1 2A 6B1 $a^{10}d^{7}$
$(C_5\times C_{15}^2):C_{12}$ 6C1 $6$ $375$ $C_3\times C_{12}$ 3E1 2A 6C-1 $a^{10}bc^{8}d^{6}$
$(C_5\times C_{15}^2):C_{12}$ 6C-1 $6$ $375$ $C_3\times C_{12}$ 3E-1 2A 6C1 $a^{2}bc^{4}d^{9}$
$(C_5\times C_{15}^2):C_{12}$ 6D1 $6$ $375$ $C_3\times C_{12}$ 3D-1 2A 6D-1 $a^{2}bc^{8}d$
$(C_5\times C_{15}^2):C_{12}$ 6D-1 $6$ $375$ $C_3\times C_{12}$ 3D1 2A 6D1 $a^{10}c^{3}d^{2}$
$(C_5\times C_{15}^2):C_{12}$ 6E1 $6$ $375$ $C_3\times C_{12}$ 3B1 2A 6E-1 $a^{6}b^{4}c^{6}d$
$(C_5\times C_{15}^2):C_{12}$ 6E-1 $6$ $375$ $C_3\times C_{12}$ 3B-1 2A 6E1 $a^{6}b^{4}c^{6}d^{11}$
$(C_5\times C_{15}^2):C_{12}$ 12A1 $12$ $125$ $C_4\times \He_3$ 6A1 4A1 12A5 $a^{3}c^{10}$
$(C_5\times C_{15}^2):C_{12}$ 12A-1 $12$ $125$ $C_4\times \He_3$ 6A-1 4A-1 12A-5 $a^{9}c^{5}$
$(C_5\times C_{15}^2):C_{12}$ 12A5 $12$ $125$ $C_4\times \He_3$ 6A-1 4A1 12A1 $a^{3}c^{5}$
$(C_5\times C_{15}^2):C_{12}$ 12A-5 $12$ $125$ $C_4\times \He_3$ 6A1 4A-1 12A-1 $a^{9}c^{10}$
$(C_5\times C_{15}^2):C_{12}$ 12B1 $12$ $375$ $C_3\times C_{12}$ 6B1 4A1 12B5 $a^{7}bc^{12}d^{4}$
$(C_5\times C_{15}^2):C_{12}$ 12B-1 $12$ $375$ $C_3\times C_{12}$ 6B-1 4A-1 12B-5 $a^{5}c^{4}d^{5}$
$(C_5\times C_{15}^2):C_{12}$ 12B5 $12$ $375$ $C_3\times C_{12}$ 6B-1 4A1 12B1 $a^{11}b^{4}c^{4}d^{2}$
$(C_5\times C_{15}^2):C_{12}$ 12B-5 $12$ $375$ $C_3\times C_{12}$ 6B1 4A-1 12B-1 $ab^{3}c^{6}d^{13}$
$(C_5\times C_{15}^2):C_{12}$ 12C1 $12$ $375$ $C_3\times C_{12}$ 6C1 4A1 12C5 $a^{11}bc^{13}d^{9}$
$(C_5\times C_{15}^2):C_{12}$ 12C-1 $12$ $375$ $C_3\times C_{12}$ 6C-1 4A-1 12C-5 $abc^{14}d^{3}$
$(C_5\times C_{15}^2):C_{12}$ 12C5 $12$ $375$ $C_3\times C_{12}$ 6C-1 4A1 12C1 $a^{7}bc^{5}$
$(C_5\times C_{15}^2):C_{12}$ 12C-5 $12$ $375$ $C_3\times C_{12}$ 6C1 4A-1 12C-1 $a^{5}bcd^{9}$
$(C_5\times C_{15}^2):C_{12}$ 12D1 $12$ $375$ $C_3\times C_{12}$ 6D1 4A1 12D5 $a^{7}c^{12}d^{2}$
$(C_5\times C_{15}^2):C_{12}$ 12D-1 $12$ $375$ $C_3\times C_{12}$ 6D-1 4A-1 12D-5 $a^{5}b^{2}c^{5}d^{10}$
$(C_5\times C_{15}^2):C_{12}$ 12D5 $12$ $375$ $C_3\times C_{12}$ 6D-1 4A1 12D1 $a^{11}b^{3}c^{2}d^{4}$
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