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Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e, f \mid b^{4}=c^{5}=d^{10}=e^{5}=f^{5}=[c,d]=[c,e]=[c,f]=[e,f]=1, a^{4}=b^{2}c^{2}e, b^{a}=b^{3}ce^{4}f^{2}, c^{a}=c^{3}d^{8}, d^{a}=c^{3}d^{7}ef^{3}, e^{a}=f^{4}, f^{a}=e^{2}, c^{b}=d^{6}, d^{b}=c^{4}d^{5}e^{2}f^{2}, e^{b}=e^{2}, f^{b}=f^{3}, e^{d}=e^{4}, f^{d}=f^{4} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 5P
$C_5^4:(C_2\times \OD_{16})$ 1A $1$ $1$ $C_5^4:(C_2\times \OD_{16})$ 1A 1A $1$
$C_5^4:(C_2\times \OD_{16})$ 2A $2$ $25$ $C_2\times C_5^2:\OD_{16}$ 1A 2A $d^{5}e$
$C_5^4:(C_2\times \OD_{16})$ 2B $2$ $25$ $C_2\times C_5^2:\OD_{16}$ 1A 2B $b^{2}cd^{9}e^{3}f^{4}$
$C_5^4:(C_2\times \OD_{16})$ 2C $2$ $50$ $D_{10}:F_5$ 1A 2C $a^{2}bc^{2}d^{6}e^{2}f^{2}$
$C_5^4:(C_2\times \OD_{16})$ 2D $2$ $50$ $D_{10}:F_5$ 1A 2D $a^{2}bc^{2}de^{3}f^{4}$
$C_5^4:(C_2\times \OD_{16})$ 2E $2$ $625$ $C_2\times \OD_{16}$ 1A 2E $b^{2}c^{4}d^{8}e^{2}f^{4}$
$C_5^4:(C_2\times \OD_{16})$ 4A1 $4$ $625$ $C_2\times \OD_{16}$ 2E 4A1 $a^{2}c^{3}d^{2}e^{4}f$
$C_5^4:(C_2\times \OD_{16})$ 4A-1 $4$ $625$ $C_2\times \OD_{16}$ 2E 4A-1 $a^{2}b^{2}cd^{6}e^{3}f^{3}$
$C_5^4:(C_2\times \OD_{16})$ 4B1 $4$ $625$ $C_2\times \OD_{16}$ 2E 4B1 $a^{2}d^{3}e^{4}f^{3}$
$C_5^4:(C_2\times \OD_{16})$ 4B-1 $4$ $625$ $C_2\times \OD_{16}$ 2E 4B-1 $a^{2}b^{2}c^{2}d^{9}e^{3}f^{2}$
$C_5^4:(C_2\times \OD_{16})$ 4C $4$ $1250$ $C_2^2\times C_4$ 2E 4C $bc^{4}d^{7}ef^{3}$
$C_5^4:(C_2\times \OD_{16})$ 4D $4$ $1250$ $C_2^2\times C_4$ 2E 4D $b^{3}c^{2}e$
$C_5^4:(C_2\times \OD_{16})$ 5A $5$ $8$ $C_5\wr C_2^2$ 5A 1A $c^{4}d^{8}$
$C_5^4:(C_2\times \OD_{16})$ 5B $5$ $8$ $C_5\wr C_2^2$ 5B 1A $f^{4}$
$C_5^4:(C_2\times \OD_{16})$ 5C $5$ $16$ $C_5^2\wr C_2$ 5C 1A $ef$
$C_5^4:(C_2\times \OD_{16})$ 5D $5$ $16$ $C_5^2\wr C_2$ 5D 1A $c^{2}$
$C_5^4:(C_2\times \OD_{16})$ 5E $5$ $16$ $C_5^2\wr C_2$ 5E 1A $c^{4}d^{8}f^{3}$
$C_5^4:(C_2\times \OD_{16})$ 5F $5$ $16$ $C_5^2\wr C_2$ 5F 1A $c^{2}d^{6}e$
$C_5^4:(C_2\times \OD_{16})$ 5G $5$ $16$ $C_5^2\wr C_2$ 5G 1A $c^{4}d^{2}f^{4}$
$C_5^4:(C_2\times \OD_{16})$ 5H $5$ $16$ $C_5^2\wr C_2$ 5H 1A $c^{3}d^{6}e^{4}$
$C_5^4:(C_2\times \OD_{16})$ 5I $5$ $32$ $C_5^4$ 5I 1A $d^{2}f$
$C_5^4:(C_2\times \OD_{16})$ 5J $5$ $32$ $C_5^4$ 5J 1A $d^{2}e$
$C_5^4:(C_2\times \OD_{16})$ 5K $5$ $32$ $C_5^4$ 5K 1A $d^{2}f^{2}$
$C_5^4:(C_2\times \OD_{16})$ 5L $5$ $32$ $C_5^4$ 5L 1A $d^{2}e^{2}$
$C_5^4:(C_2\times \OD_{16})$ 5M $5$ $32$ $C_5^4$ 5M 1A $d^{2}ef$
$C_5^4:(C_2\times \OD_{16})$ 5N $5$ $32$ $C_5^4$ 5N 1A $cef$
$C_5^4:(C_2\times \OD_{16})$ 5O $5$ $32$ $C_5^4$ 5O 1A $d^{2}ef^{2}$
$C_5^4:(C_2\times \OD_{16})$ 5P $5$ $32$ $C_5^4$ 5P 1A $d^{2}e^{2}f$
$C_5^4:(C_2\times \OD_{16})$ 5Q $5$ $32$ $C_5^4$ 5Q 1A $d^{4}ef$
$C_5^4:(C_2\times \OD_{16})$ 5R $5$ $32$ $C_5^4$ 5R 1A $cef^{2}$
$C_5^4:(C_2\times \OD_{16})$ 5S $5$ $32$ $C_5^4$ 5S 1A $ce^{2}f$
$C_5^4:(C_2\times \OD_{16})$ 5T $5$ $32$ $C_5^4$ 5T 1A $c^{2}ef$
$C_5^4:(C_2\times \OD_{16})$ 5U $5$ $32$ $C_5^4$ 5U 1A $cd^{2}ef$
$C_5^4:(C_2\times \OD_{16})$ 5V $5$ $32$ $C_5^4$ 5V 1A $cd^{2}ef^{2}$
$C_5^4:(C_2\times \OD_{16})$ 5W $5$ $32$ $C_5^4$ 5W 1A $cd^{2}e^{2}f$
$C_5^4:(C_2\times \OD_{16})$ 5X $5$ $32$ $C_5^4$ 5X 1A $cd^{2}e^{2}f^{2}$
$C_5^4:(C_2\times \OD_{16})$ 8A1 $8$ $1250$ $C_2\times C_8$ 4A1 8A1 $ab^{3}cf^{3}$
$C_5^4:(C_2\times \OD_{16})$ 8A-1 $8$ $1250$ $C_2\times C_8$ 4A-1 8A-1 $a^{3}bd^{8}ef^{4}$
$C_5^4:(C_2\times \OD_{16})$ 8B1 $8$ $1250$ $C_2\times C_8$ 4A1 8B1 $ab^{2}c^{4}d^{5}e^{3}f$
$C_5^4:(C_2\times \OD_{16})$ 8B-1 $8$ $1250$ $C_2\times C_8$ 4A-1 8B-1 $a^{3}c^{2}d^{7}ef^{4}$
$C_5^4:(C_2\times \OD_{16})$ 8C1 $8$ $1250$ $C_2\times C_8$ 4A1 8C1 $ab^{2}c^{2}d^{4}e^{2}f$
$C_5^4:(C_2\times \OD_{16})$ 8C-1 $8$ $1250$ $C_2\times C_8$ 4A-1 8C-1 $a^{3}c^{4}d^{6}e^{4}f^{3}$
$C_5^4:(C_2\times \OD_{16})$ 8D1 $8$ $1250$ $C_2\times C_8$ 4A-1 8D1 $a^{3}b^{3}cd^{3}e^{4}f^{2}$
$C_5^4:(C_2\times \OD_{16})$ 8D-1 $8$ $1250$ $C_2\times C_8$ 4A1 8D-1 $abd^{3}e^{4}f^{2}$
$C_5^4:(C_2\times \OD_{16})$ 10A $10$ $200$ $C_5\times D_{10}$ 5A 2A $c^{2}d^{9}e$
$C_5^4:(C_2\times \OD_{16})$ 10B $10$ $200$ $C_5\times D_{10}$ 5B 2B $b^{2}cd^{9}e^{3}f$
$C_5^4:(C_2\times \OD_{16})$ 10C $10$ $200$ $C_5\times D_{10}$ 5B 2C $a^{2}bc^{2}d^{6}f^{2}$
$C_5^4:(C_2\times \OD_{16})$ 10D $10$ $200$ $C_5\times D_{10}$ 5A 2D $a^{2}bcd^{3}e^{3}f^{4}$
$C_5^4:(C_2\times \OD_{16})$ 10E $10$ $200$ $C_5\times D_{10}$ 5B 2D $a^{2}b^{3}d^{3}e^{2}$
$C_5^4:(C_2\times \OD_{16})$ 10F $10$ $200$ $C_5\times D_{10}$ 5A 2C $a^{2}bc^{3}d^{2}e^{2}$
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