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Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e \mid b^{15}=c^{5}=d^{5}=e^{5}=[b,e]=[c,d]=[c,e]=[d,e]=1, a^{6}=cd^{4}, b^{a}=b^{5}de^{3}, c^{a}=cd^{3}, d^{a}=d^{4}, e^{a}=b^{3}e^{4}, c^{b}=c^{4}d^{2}, d^{b}=c^{2} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 5P
$C_5^4:(C_3\times S_3)$ 1A $1$ $1$ $C_5^4:(C_3\times S_3)$ 1A 1A 1A $1$
$C_5^4:(C_3\times S_3)$ 2A $2$ $15$ $C_5^2:C_{30}$ 1A 2A 2A $a^{3}c^{2}d^{3}$
$C_5^4:(C_3\times S_3)$ 3A1 $3$ $25$ $C_3\times C_5^2:S_3$ 3A-1 1A 3A-1 $a^{4}cd^{4}e^{3}$
$C_5^4:(C_3\times S_3)$ 3A-1 $3$ $25$ $C_3\times C_5^2:S_3$ 3A1 1A 3A1 $a^{2}b^{9}c^{3}d^{2}$
$C_5^4:(C_3\times S_3)$ 3B $3$ $50$ $C_5^2:C_3^2$ 3B 1A 3B $b^{5}c^{3}$
$C_5^4:(C_3\times S_3)$ 3C1 $3$ $1250$ $C_3^2$ 3C-1 1A 3C-1 $a^{2}bcd^{2}e$
$C_5^4:(C_3\times S_3)$ 3C-1 $3$ $1250$ $C_3^2$ 3C1 1A 3C1 $a^{4}b^{2}c^{4}e^{3}$
$C_5^4:(C_3\times S_3)$ 5A1 $5$ $3$ $C_5^2\times C_5^2:S_3$ 5A2 5A-2 1A $b^{12}$
$C_5^4:(C_3\times S_3)$ 5A-1 $5$ $3$ $C_5^2\times C_5^2:S_3$ 5A-2 5A2 1A $b^{3}$
$C_5^4:(C_3\times S_3)$ 5A2 $5$ $3$ $C_5^2\times C_5^2:S_3$ 5A-1 5A1 1A $b^{9}$
$C_5^4:(C_3\times S_3)$ 5A-2 $5$ $3$ $C_5^2\times C_5^2:S_3$ 5A1 5A-1 1A $b^{6}$
$C_5^4:(C_3\times S_3)$ 5B1 $5$ $3$ $C_5^2\times C_5^2:S_3$ 5B2 5B-2 1A $b^{12}e^{2}$
$C_5^4:(C_3\times S_3)$ 5B-1 $5$ $3$ $C_5^2\times C_5^2:S_3$ 5B-2 5B2 1A $b^{3}e^{3}$
$C_5^4:(C_3\times S_3)$ 5B2 $5$ $3$ $C_5^2\times C_5^2:S_3$ 5B-1 5B1 1A $b^{9}e^{4}$
$C_5^4:(C_3\times S_3)$ 5B-2 $5$ $3$ $C_5^2\times C_5^2:S_3$ 5B1 5B-1 1A $b^{6}e$
$C_5^4:(C_3\times S_3)$ 5C1 $5$ $3$ $C_5^3:C_{30}$ 5C2 5C-2 1A $c^{2}d^{2}$
$C_5^4:(C_3\times S_3)$ 5C-1 $5$ $3$ $C_5^3:C_{30}$ 5C-2 5C2 1A $c^{3}d^{3}$
$C_5^4:(C_3\times S_3)$ 5C2 $5$ $3$ $C_5^3:C_{30}$ 5C-1 5C1 1A $c^{4}d^{4}$
$C_5^4:(C_3\times S_3)$ 5C-2 $5$ $3$ $C_5^3:C_{30}$ 5C1 5C-1 1A $cd$
$C_5^4:(C_3\times S_3)$ 5D1 $5$ $6$ $C_5^3:C_{15}$ 5D2 5D2 1A $c^{4}$
$C_5^4:(C_3\times S_3)$ 5D2 $5$ $6$ $C_5^3:C_{15}$ 5D1 5D1 1A $c^{3}$
$C_5^4:(C_3\times S_3)$ 5E1 $5$ $9$ $D_5\times C_5^3$ 5E2 5E-2 1A $cd^{4}e^{2}$
$C_5^4:(C_3\times S_3)$ 5E-1 $5$ $9$ $D_5\times C_5^3$ 5E-2 5E2 1A $c^{4}de^{3}$
$C_5^4:(C_3\times S_3)$ 5E2 $5$ $9$ $D_5\times C_5^3$ 5E-1 5E1 1A $c^{2}d^{3}e^{4}$
$C_5^4:(C_3\times S_3)$ 5E-2 $5$ $9$ $D_5\times C_5^3$ 5E1 5E-1 1A $c^{3}d^{2}e$
$C_5^4:(C_3\times S_3)$ 5F1 $5$ $9$ $D_5\times C_5^3$ 5F2 5F-2 1A $b^{6}cd^{4}$
$C_5^4:(C_3\times S_3)$ 5F-1 $5$ $9$ $D_5\times C_5^3$ 5F-2 5F2 1A $b^{9}c^{4}d$
$C_5^4:(C_3\times S_3)$ 5F2 $5$ $9$ $D_5\times C_5^3$ 5F-1 5F1 1A $b^{12}c^{2}d^{3}$
$C_5^4:(C_3\times S_3)$ 5F-2 $5$ $9$ $D_5\times C_5^3$ 5F1 5F-1 1A $b^{3}c^{3}d^{2}$
$C_5^4:(C_3\times S_3)$ 5G1 $5$ $9$ $D_5\times C_5^3$ 5G2 5G-2 1A $cd^{4}e^{4}$
$C_5^4:(C_3\times S_3)$ 5G-1 $5$ $9$ $D_5\times C_5^3$ 5G-2 5G2 1A $c^{4}de$
$C_5^4:(C_3\times S_3)$ 5G2 $5$ $9$ $D_5\times C_5^3$ 5G-1 5G1 1A $c^{2}d^{3}e^{3}$
$C_5^4:(C_3\times S_3)$ 5G-2 $5$ $9$ $D_5\times C_5^3$ 5G1 5G-1 1A $c^{3}d^{2}e^{2}$
$C_5^4:(C_3\times S_3)$ 5H1 $5$ $9$ $D_5\times C_5^3$ 5H2 5H-2 1A $b^{12}cd^{4}$
$C_5^4:(C_3\times S_3)$ 5H-1 $5$ $9$ $D_5\times C_5^3$ 5H-2 5H2 1A $b^{3}c^{4}d$
$C_5^4:(C_3\times S_3)$ 5H2 $5$ $9$ $D_5\times C_5^3$ 5H-1 5H1 1A $b^{9}c^{2}d^{3}$
$C_5^4:(C_3\times S_3)$ 5H-2 $5$ $9$ $D_5\times C_5^3$ 5H1 5H-1 1A $b^{6}c^{3}d^{2}$
$C_5^4:(C_3\times S_3)$ 5I1 $5$ $9$ $D_5\times C_5^3$ 5I2 5I-2 1A $b^{6}cd^{4}e^{2}$
$C_5^4:(C_3\times S_3)$ 5I-1 $5$ $9$ $D_5\times C_5^3$ 5I-2 5I2 1A $b^{9}c^{4}de^{3}$
$C_5^4:(C_3\times S_3)$ 5I2 $5$ $9$ $D_5\times C_5^3$ 5I-1 5I1 1A $b^{12}c^{2}d^{3}e^{4}$
$C_5^4:(C_3\times S_3)$ 5I-2 $5$ $9$ $D_5\times C_5^3$ 5I1 5I-1 1A $b^{3}c^{3}d^{2}e$
$C_5^4:(C_3\times S_3)$ 5J1 $5$ $9$ $D_5\times C_5^3$ 5J2 5J-2 1A $b^{6}cd^{4}e^{4}$
$C_5^4:(C_3\times S_3)$ 5J-1 $5$ $9$ $D_5\times C_5^3$ 5J-2 5J2 1A $b^{9}c^{4}de$
$C_5^4:(C_3\times S_3)$ 5J2 $5$ $9$ $D_5\times C_5^3$ 5J-1 5J1 1A $b^{12}c^{2}d^{3}e^{3}$
$C_5^4:(C_3\times S_3)$ 5J-2 $5$ $9$ $D_5\times C_5^3$ 5J1 5J-1 1A $b^{3}c^{3}d^{2}e^{2}$
$C_5^4:(C_3\times S_3)$ 5K1 $5$ $9$ $D_5\times C_5^3$ 5K2 5K-2 1A $b^{12}cd^{4}e^{3}$
$C_5^4:(C_3\times S_3)$ 5K-1 $5$ $9$ $D_5\times C_5^3$ 5K-2 5K2 1A $b^{3}c^{4}de^{2}$
$C_5^4:(C_3\times S_3)$ 5K2 $5$ $9$ $D_5\times C_5^3$ 5K-1 5K1 1A $b^{9}c^{2}d^{3}e$
$C_5^4:(C_3\times S_3)$ 5K-2 $5$ $9$ $D_5\times C_5^3$ 5K1 5K-1 1A $b^{6}c^{3}d^{2}e^{4}$
$C_5^4:(C_3\times S_3)$ 5L1 $5$ $9$ $D_5\times C_5^3$ 5L2 5L-2 1A $b^{3}cd^{4}e$
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