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

Group Label Order Size Centralizer Powers Representative
2P 3P 5P
$(C_5\times C_{15}^2):C_6$ 1A $1$ $1$ $(C_5\times C_{15}^2):C_6$ 1A 1A 1A $1$
$(C_5\times C_{15}^2):C_6$ 2A $2$ $125$ $C_2\times \He_3$ 1A 2A 2A $a^{3}$
$(C_5\times C_{15}^2):C_6$ 3A1 $3$ $1$ $(C_5\times C_{15}^2):C_6$ 3A-1 1A 3A-1 $c^{10}$
$(C_5\times C_{15}^2):C_6$ 3A-1 $3$ $1$ $(C_5\times C_{15}^2):C_6$ 3A1 1A 3A1 $c^{5}$
$(C_5\times C_{15}^2):C_6$ 3B1 $3$ $3$ $C_3^2\times C_5^2:D_5$ 3B-1 1A 3B-1 $d^{5}$
$(C_5\times C_{15}^2):C_6$ 3B-1 $3$ $3$ $C_3^2\times C_5^2:D_5$ 3B1 1A 3B1 $d^{10}$
$(C_5\times C_{15}^2):C_6$ 3C1 $3$ $75$ $C_3^2\times D_5$ 3C-1 1A 3C-1 $a^{4}b^{2}c^{12}d^{4}$
$(C_5\times C_{15}^2):C_6$ 3C-1 $3$ $75$ $C_3^2\times D_5$ 3C1 1A 3C1 $a^{2}bc^{4}d^{14}$
$(C_5\times C_{15}^2):C_6$ 3D1 $3$ $75$ $C_3^2\times D_5$ 3D-1 1A 3D-1 $a^{4}b^{2}c^{8}d^{8}$
$(C_5\times C_{15}^2):C_6$ 3D-1 $3$ $75$ $C_3^2\times D_5$ 3D1 1A 3D1 $a^{2}b^{3}c^{9}d^{10}$
$(C_5\times C_{15}^2):C_6$ 3E1 $3$ $75$ $C_3^2\times D_5$ 3E-1 1A 3E-1 $a^{2}b^{4}c^{10}d^{12}$
$(C_5\times C_{15}^2):C_6$ 3E-1 $3$ $75$ $C_3^2\times D_5$ 3E1 1A 3E1 $a^{4}c^{2}d^{3}$
$(C_5\times C_{15}^2):C_6$ 5A1 $5$ $2$ $C_{15}^2:C_{15}$ 5A2 5A2 1A $b^{4}c^{6}d^{12}$
$(C_5\times C_{15}^2):C_6$ 5A2 $5$ $2$ $C_{15}^2:C_{15}$ 5A1 5A1 1A $b^{3}c^{12}d^{9}$
$(C_5\times C_{15}^2):C_6$ 5B1 $5$ $6$ $C_5\times C_{15}^2$ 5B2 5B2 1A $d^{3}$
$(C_5\times C_{15}^2):C_6$ 5B2 $5$ $6$ $C_5\times C_{15}^2$ 5B1 5B1 1A $d^{6}$
$(C_5\times C_{15}^2):C_6$ 5C1 $5$ $6$ $C_5\times C_{15}^2$ 5C2 5C2 1A $c^{3}$
$(C_5\times C_{15}^2):C_6$ 5C2 $5$ $6$ $C_5\times C_{15}^2$ 5C1 5C1 1A $c^{6}$
$(C_5\times C_{15}^2):C_6$ 5D1 $5$ $6$ $C_5\times C_{15}^2$ 5D2 5D2 1A $c^{9}d^{3}$
$(C_5\times C_{15}^2):C_6$ 5D2 $5$ $6$ $C_5\times C_{15}^2$ 5D1 5D1 1A $c^{3}d^{6}$
$(C_5\times C_{15}^2):C_6$ 5E1 $5$ $6$ $C_5\times C_{15}^2$ 5E2 5E2 1A $c^{3}d^{3}$
$(C_5\times C_{15}^2):C_6$ 5E2 $5$ $6$ $C_5\times C_{15}^2$ 5E1 5E1 1A $c^{6}d^{6}$
$(C_5\times C_{15}^2):C_6$ 5F1 $5$ $6$ $C_5\times C_{15}^2$ 5F2 5F2 1A $b^{3}d^{3}$
$(C_5\times C_{15}^2):C_6$ 5F2 $5$ $6$ $C_5\times C_{15}^2$ 5F1 5F1 1A $bd^{6}$
$(C_5\times C_{15}^2):C_6$ 5G1 $5$ $6$ $C_5\times C_{15}^2$ 5G2 5G2 1A $b^{3}c^{3}$
$(C_5\times C_{15}^2):C_6$ 5G2 $5$ $6$ $C_5\times C_{15}^2$ 5G1 5G1 1A $bc^{6}$
$(C_5\times C_{15}^2):C_6$ 5H1 $5$ $6$ $C_5\times C_{15}^2$ 5H2 5H2 1A $c^{9}d^{6}$
$(C_5\times C_{15}^2):C_6$ 5H2 $5$ $6$ $C_5\times C_{15}^2$ 5H1 5H1 1A $c^{3}d^{12}$
$(C_5\times C_{15}^2):C_6$ 5I1 $5$ $6$ $C_5\times C_{15}^2$ 5I2 5I2 1A $bc^{3}$
$(C_5\times C_{15}^2):C_6$ 5I2 $5$ $6$ $C_5\times C_{15}^2$ 5I1 5I1 1A $b^{2}c^{6}$
$(C_5\times C_{15}^2):C_6$ 5J1 $5$ $6$ $C_5\times C_{15}^2$ 5J2 5J2 1A $b^{4}d^{6}$
$(C_5\times C_{15}^2):C_6$ 5J2 $5$ $6$ $C_5\times C_{15}^2$ 5J1 5J1 1A $b^{3}d^{12}$
$(C_5\times C_{15}^2):C_6$ 5K1 $5$ $6$ $C_5\times C_{15}^2$ 5K2 5K2 1A $b^{3}c^{9}d^{3}$
$(C_5\times C_{15}^2):C_6$ 5K2 $5$ $6$ $C_5\times C_{15}^2$ 5K1 5K1 1A $bc^{3}d^{6}$
$(C_5\times C_{15}^2):C_6$ 6A1 $6$ $125$ $C_2\times \He_3$ 3A1 2A 6A-1 $a^{3}bc^{14}$
$(C_5\times C_{15}^2):C_6$ 6A-1 $6$ $125$ $C_2\times \He_3$ 3A-1 2A 6A1 $a^{3}bc^{4}$
$(C_5\times C_{15}^2):C_6$ 6B1 $6$ $375$ $C_3\times C_6$ 3B1 2A 6B-1 $a^{3}d^{10}$
$(C_5\times C_{15}^2):C_6$ 6B-1 $6$ $375$ $C_3\times C_6$ 3B-1 2A 6B1 $a^{3}d^{5}$
$(C_5\times C_{15}^2):C_6$ 6C1 $6$ $375$ $C_3\times C_6$ 3D1 2A 6C-1 $a^{5}c^{6}d$
$(C_5\times C_{15}^2):C_6$ 6C-1 $6$ $375$ $C_3\times C_6$ 3D-1 2A 6C1 $abc^{2}d^{8}$
$(C_5\times C_{15}^2):C_6$ 6D1 $6$ $375$ $C_3\times C_6$ 3E1 2A 6D-1 $ac^{2}d^{9}$
$(C_5\times C_{15}^2):C_6$ 6D-1 $6$ $375$ $C_3\times C_6$ 3E-1 2A 6D1 $a^{5}bc^{4}$
$(C_5\times C_{15}^2):C_6$ 6E1 $6$ $375$ $C_3\times C_6$ 3C1 2A 6E-1 $a^{5}bc^{12}d^{14}$
$(C_5\times C_{15}^2):C_6$ 6E-1 $6$ $375$ $C_3\times C_6$ 3C-1 2A 6E1 $ac^{7}d$
$(C_5\times C_{15}^2):C_6$ 15A1 $15$ $2$ $C_{15}^2:C_{15}$ 15A2 5A1 3A1 $b^{3}c^{2}d^{9}$
$(C_5\times C_{15}^2):C_6$ 15A-1 $15$ $2$ $C_{15}^2:C_{15}$ 15A-2 5A1 3A-1 $b^{2}c^{13}d^{6}$
$(C_5\times C_{15}^2):C_6$ 15A2 $15$ $2$ $C_{15}^2:C_{15}$ 15A1 5A2 3A-1 $b^{4}cd^{12}$
$(C_5\times C_{15}^2):C_6$ 15A-2 $15$ $2$ $C_{15}^2:C_{15}$ 15A-1 5A2 3A1 $bc^{14}d^{3}$
$(C_5\times C_{15}^2):C_6$ 15B1 $15$ $6$ $C_5\times C_{15}^2$ 15B2 5B1 3B1 $d$
$(C_5\times C_{15}^2):C_6$ 15B-1 $15$ $6$ $C_5\times C_{15}^2$ 15B-2 5B1 3B-1 $d^{11}$
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