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Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e, f \mid a^{4}=c^{2}=d^{30}=e^{15}=f^{15}=[c,e]=[e,f]=1, b^{3}=d^{18}e^{3}f^{9}, b^{a}=b^{2}cd^{2}f^{8}, c^{a}=cd^{6}f^{14}, d^{a}=cd^{21}ef, e^{a}=e^{2}, f^{a}=d^{26}, c^{b}=cd^{7}e^{9}, d^{b}=cd^{22}e^{11}f^{10}, e^{b}=f^{13}, f^{b}=d^{2}, d^{c}=d^{29}, f^{c}=f^{14}, e^{d}=e^{14}, f^{d}=f^{14} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 5P
$(C_{15}^3.A_4):C_4$ 1A $1$ $1$ $(C_{15}^3.A_4):C_4$ 1A 1A 1A $1$
$(C_{15}^3.A_4):C_4$ 2A $2$ $125$ $(C_3^2\times C_6).S_4$ 1A 2A 2A $a^{2}cd^{8}e^{3}f^{10}$
$(C_{15}^3.A_4):C_4$ 2B $2$ $135$ $(C_5\times C_{10}).D_{12}$ 1A 2B 2B $a^{2}d^{20}e^{6}f^{10}$
$(C_{15}^3.A_4):C_4$ 2C $2$ $675$ $D_{30}:C_4$ 1A 2C 2C $cd^{2}f^{10}$
$(C_{15}^3.A_4):C_4$ 3A1 $3$ $4$ $C_{15}\wr C_3:C_4$ 3A-1 1A 3A-1 $d^{10}e^{5}f^{5}$
$(C_{15}^3.A_4):C_4$ 3A-1 $3$ $4$ $C_{15}\wr C_3:C_4$ 3A1 1A 3A1 $d^{20}e^{10}f^{10}$
$(C_{15}^3.A_4):C_4$ 3B $3$ $6$ $C_3\times C_5^3.(C_2.S_3^2)$ 3B 1A 3B $f^{5}$
$(C_{15}^3.A_4):C_4$ 3C $3$ $12$ $C_3\times C_5^3:(C_3\times C_3:C_4)$ 3C 1A 3C $d^{10}f^{5}$
$(C_{15}^3.A_4):C_4$ 3D $3$ $1800$ $C_3^2\times D_5$ 3D 1A 3D $b^{2}d^{3}e^{7}f^{5}$
$(C_{15}^3.A_4):C_4$ 4A1 $4$ $2250$ $S_3\times C_{12}$ 2A 4A-1 4A1 $ab^{2}d^{24}e^{3}f^{7}$
$(C_{15}^3.A_4):C_4$ 4A-1 $4$ $2250$ $S_3\times C_{12}$ 2A 4A1 4A-1 $a^{3}b^{2}d^{3}e^{3}f^{6}$
$(C_{15}^3.A_4):C_4$ 4B1 $4$ $4050$ $C_4\times D_5$ 2B 4B-1 4B1 $a^{3}cd^{20}e^{14}f^{14}$
$(C_{15}^3.A_4):C_4$ 4B-1 $4$ $4050$ $C_4\times D_5$ 2B 4B1 4B-1 $acd^{10}e^{2}f^{4}$
$(C_{15}^3.A_4):C_4$ 5A $5$ $12$ $C_{15}^3.C_2^2$ 5A 5A 1A $d^{18}$
$(C_{15}^3.A_4):C_4$ 5B1 $5$ $12$ $(C_3\times C_{15}^2).C_{20}$ 5B2 5B2 1A $e^{3}f^{3}$
$(C_{15}^3.A_4):C_4$ 5B2 $5$ $12$ $(C_3\times C_{15}^2).C_{20}$ 5B1 5B1 1A $e^{6}f^{6}$
$(C_{15}^3.A_4):C_4$ 5C $5$ $16$ $C_{15}\wr C_3$ 5C 5C 1A $d^{18}e^{3}f^{9}$
$(C_{15}^3.A_4):C_4$ 5D $5$ $24$ $C_5\times C_{15}\times C_3:D_{15}$ 5D 5D 1A $d^{6}f^{3}$
$(C_{15}^3.A_4):C_4$ 5E $5$ $48$ $C_{15}^3$ 5E 5E 1A $d^{6}e^{3}f^{3}$
$(C_{15}^3.A_4):C_4$ 6A $6$ $270$ $C_{30}:F_5$ 3B 2B 6A $a^{2}d^{5}e^{9}f^{14}$
$(C_{15}^3.A_4):C_4$ 6B1 $6$ $500$ $\He_3:C_{12}$ 3A-1 2A 6B-1 $a^{2}cd^{28}e^{13}$
$(C_{15}^3.A_4):C_4$ 6B-1 $6$ $500$ $\He_3:C_{12}$ 3A1 2A 6B1 $a^{2}cd^{18}e^{8}f^{5}$
$(C_{15}^3.A_4):C_4$ 6C $6$ $750$ $C_6.S_3^2$ 3B 2A 6C $a^{2}cd^{8}e^{2}f$
$(C_{15}^3.A_4):C_4$ 6D $6$ $1350$ $C_6\times F_5$ 3B 2C 6D $cd^{2}e^{5}f^{10}$
$(C_{15}^3.A_4):C_4$ 6E $6$ $1500$ $C_3^2:C_{12}$ 3C 2A 6E $a^{2}cd^{16}e^{6}f^{14}$
$(C_{15}^3.A_4):C_4$ 6F $6$ $9000$ $C_3\times C_6$ 3D 2A 6F $a^{2}b^{2}d^{18}e^{3}f^{8}$
$(C_{15}^3.A_4):C_4$ 9A1 $9$ $1800$ $C_9\times D_5$ 9A-1 3A-1 9A-1 $b^{2}de^{14}$
$(C_{15}^3.A_4):C_4$ 9A-1 $9$ $1800$ $C_9\times D_5$ 9A1 3A1 9A1 $bcd^{4}ef^{3}$
$(C_{15}^3.A_4):C_4$ 10A1 $10$ $540$ $C_{15}:C_{20}$ 5B2 10A3 2B $a^{2}d^{14}e^{6}f$
$(C_{15}^3.A_4):C_4$ 10A3 $10$ $540$ $C_{15}:C_{20}$ 5B1 10A1 2B $a^{2}d^{2}e^{6}f^{13}$
$(C_{15}^3.A_4):C_4$ 10B1 $10$ $540$ $C_5\times D_{30}$ 5A 10B3 2B $a^{2}cd^{29}e^{9}f^{7}$
$(C_{15}^3.A_4):C_4$ 10B3 $10$ $540$ $C_5\times D_{30}$ 5A 10B1 2B $a^{2}cd^{29}e^{9}f$
$(C_{15}^3.A_4):C_4$ 10C $10$ $1080$ $C_5\times C_{30}$ 5D 10C 2B $a^{2}d^{19}e^{10}f$
$(C_{15}^3.A_4):C_4$ 10D $10$ $2700$ $S_3\times C_{10}$ 5A 10D 2C $cd^{2}e^{6}f^{10}$
$(C_{15}^3.A_4):C_4$ 12A1 $12$ $2250$ $S_3\times C_{12}$ 6C 4A-1 12A5 $ad^{3}e^{12}f^{4}$
$(C_{15}^3.A_4):C_4$ 12A-1 $12$ $2250$ $S_3\times C_{12}$ 6C 4A1 12A-5 $a^{3}cd^{5}e^{13}f^{12}$
$(C_{15}^3.A_4):C_4$ 12A5 $12$ $2250$ $S_3\times C_{12}$ 6C 4A-1 12A1 $ad^{3}e^{7}f^{4}$
$(C_{15}^3.A_4):C_4$ 12A-5 $12$ $2250$ $S_3\times C_{12}$ 6C 4A1 12A-1 $a^{3}cd^{5}e^{3}f^{12}$
$(C_{15}^3.A_4):C_4$ 12B1 $12$ $4500$ $C_3\times C_{12}$ 6E 4A-1 12B1 $ad^{23}e^{8}f^{7}$
$(C_{15}^3.A_4):C_4$ 12B-1 $12$ $4500$ $C_3\times C_{12}$ 6E 4A1 12B-1 $a^{3}cd^{23}e^{11}f^{2}$
$(C_{15}^3.A_4):C_4$ 12C1 $12$ $4500$ $C_3\times C_{12}$ 6B1 4A1 12C5 $a^{3}b^{2}d^{23}e^{8}f^{11}$
$(C_{15}^3.A_4):C_4$ 12C-1 $12$ $4500$ $C_3\times C_{12}$ 6B-1 4A-1 12C-5 $ab^{2}d^{4}e^{13}f^{2}$
$(C_{15}^3.A_4):C_4$ 12C5 $12$ $4500$ $C_3\times C_{12}$ 6B-1 4A1 12C1 $a^{3}b^{2}d^{13}e^{13}f$
$(C_{15}^3.A_4):C_4$ 12C-5 $12$ $4500$ $C_3\times C_{12}$ 6B1 4A-1 12C-1 $ab^{2}d^{14}e^{8}f^{12}$
$(C_{15}^3.A_4):C_4$ 15A1 $15$ $16$ $C_{15}\wr C_3$ 15A-1 5C 3A-1 $d^{14}e^{4}f^{8}$
$(C_{15}^3.A_4):C_4$ 15A-1 $15$ $16$ $C_{15}\wr C_3$ 15A1 5C 3A1 $d^{16}e^{11}f^{7}$
$(C_{15}^3.A_4):C_4$ 15B $15$ $24$ $C_3\times C_5^3:(C_3:S_3)$ 15B 5A 3B $e^{14}$
$(C_{15}^3.A_4):C_4$ 15C1 $15$ $24$ $C_5\times C_{15}\times C_3:D_{15}$ 15C2 5B1 3B $d^{24}e^{12}f^{10}$
$(C_{15}^3.A_4):C_4$ 15C2 $15$ $24$ $C_5\times C_{15}\times C_3:D_{15}$ 15C1 5B2 3B $d^{18}e^{9}f^{5}$
$(C_{15}^3.A_4):C_4$ 15D1 $15$ $24$ $C_5\times C_{15}\times C_3:D_{15}$ 15D2 5A 3B $d^{20}f^{3}$
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