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

Group Label Order Size Centralizer Powers Representative
2P 5P
$D_5^3.C_2^2$ 1A $1$ $1$ $D_5^3.C_2^2$ 1A 1A $1$
$D_5^3.C_2^2$ 2A $2$ $5$ $C_4\times D_5\wr C_2$ 1A 2A $a^{2}b^{4}de^{2}$
$D_5^3.C_2^2$ 2B $2$ $10$ $D_{10}\times F_5$ 1A 2B $b^{5}$
$D_5^3.C_2^2$ 2C $2$ $10$ $D_{10}\times F_5$ 1A 2C $b^{5}cd^{4}e$
$D_5^3.C_2^2$ 2D $2$ $25$ $D_4\times F_5$ 1A 2D $c^{2}d^{2}e^{4}$
$D_5^3.C_2^2$ 2E $2$ $50$ $C_4\times D_{10}$ 1A 2E $a^{2}bc^{3}e^{2}$
$D_5^3.C_2^2$ 2F $2$ $50$ $C_4\times D_{10}$ 1A 2F $a^{2}b^{5}c^{2}d^{4}e^{2}$
$D_5^3.C_2^2$ 2G $2$ $125$ $C_4\times D_4$ 1A 2G $a^{2}b^{8}c^{2}d^{3}e^{3}$
$D_5^3.C_2^2$ 4A1 $4$ $5$ $C_4\times D_5\wr C_2$ 2A 4A1 $abc^{2}de^{4}$
$D_5^3.C_2^2$ 4A-1 $4$ $5$ $C_4\times D_5\wr C_2$ 2A 4A-1 $a^{3}b^{3}c^{2}d^{2}e^{3}$
$D_5^3.C_2^2$ 4B $4$ $50$ $C_4\times F_5$ 2D 4B $c^{3}d^{4}e^{3}$
$D_5^3.C_2^2$ 4C1 $4$ $50$ $C_4\times D_{10}$ 2A 4C1 $ac^{3}d^{2}e$
$D_5^3.C_2^2$ 4C-1 $4$ $50$ $C_4\times D_{10}$ 2A 4C-1 $a^{3}c^{3}d^{2}$
$D_5^3.C_2^2$ 4D1 $4$ $50$ $C_4\times D_{10}$ 2A 4D1 $a^{3}b^{6}d^{4}$
$D_5^3.C_2^2$ 4D-1 $4$ $50$ $C_4\times D_{10}$ 2A 4D-1 $ab^{2}d^{3}e^{3}$
$D_5^3.C_2^2$ 4E1 $4$ $125$ $C_4\times D_4$ 2A 4E1 $ab^{3}d^{4}e^{4}$
$D_5^3.C_2^2$ 4E-1 $4$ $125$ $C_4\times D_4$ 2A 4E-1 $a^{3}b^{9}d^{3}e^{2}$
$D_5^3.C_2^2$ 4F $4$ $250$ $C_4^2$ 2D 4F $a^{2}b^{2}cd^{3}e^{2}$
$D_5^3.C_2^2$ 4G1 $4$ $250$ $C_4^2$ 2G 4G1 $ab^{7}cd^{2}e^{2}$
$D_5^3.C_2^2$ 4G-1 $4$ $250$ $C_4^2$ 2G 4G-1 $a^{3}bc^{3}d^{4}e$
$D_5^3.C_2^2$ 5A $5$ $4$ $D_5^2:C_{10}$ 5A 1A $b^{2}e$
$D_5^3.C_2^2$ 5B1 $5$ $4$ $D_5^2.C_{10}$ 5B2 1A $e^{2}$
$D_5^3.C_2^2$ 5B2 $5$ $4$ $D_5^2.C_{10}$ 5B1 1A $e^{4}$
$D_5^3.C_2^2$ 5C1 $5$ $4$ $D_5^2.C_{10}$ 5C2 1A $d^{2}e^{3}$
$D_5^3.C_2^2$ 5C2 $5$ $4$ $D_5^2.C_{10}$ 5C1 1A $d^{4}e$
$D_5^3.C_2^2$ 5D $5$ $8$ $F_5\times C_5^2$ 5D 1A $de^{3}$
$D_5^3.C_2^2$ 5E1 $5$ $16$ $D_5\times C_5^2$ 5E2 1A $b^{4}d^{2}$
$D_5^3.C_2^2$ 5E2 $5$ $16$ $D_5\times C_5^2$ 5E1 1A $b^{8}d^{4}$
$D_5^3.C_2^2$ 5F1 $5$ $16$ $D_5\times C_5^2$ 5F2 1A $b^{8}d^{3}e^{4}$
$D_5^3.C_2^2$ 5F2 $5$ $16$ $D_5\times C_5^2$ 5F1 1A $b^{6}de^{3}$
$D_5^3.C_2^2$ 5G $5$ $32$ $C_5^3$ 5G 1A $b^{2}d^{2}$
$D_5^3.C_2^2$ 10A1 $10$ $20$ $C_{10}\times F_5$ 5B1 2B $b^{5}e$
$D_5^3.C_2^2$ 10A3 $10$ $20$ $C_{10}\times F_5$ 5B2 2B $b^{5}c^{2}$
$D_5^3.C_2^2$ 10B1 $10$ $20$ $C_{10}\times F_5$ 5C1 2C $b^{5}c$
$D_5^3.C_2^2$ 10B3 $10$ $20$ $C_{10}\times F_5$ 5C2 2C $b^{5}ce$
$D_5^3.C_2^2$ 10C1 $10$ $20$ $D_5\times C_{20}$ 5C2 2A $a^{2}b^{4}d^{4}e^{4}$
$D_5^3.C_2^2$ 10C3 $10$ $20$ $D_5\times C_{20}$ 5C1 2A $a^{2}b^{4}e^{3}$
$D_5^3.C_2^2$ 10D1 $10$ $20$ $D_5\times C_{20}$ 5B2 2A $a^{2}b^{4}de^{4}$
$D_5^3.C_2^2$ 10D3 $10$ $20$ $D_5\times C_{20}$ 5B1 2A $a^{2}b^{4}de^{3}$
$D_5^3.C_2^2$ 10E $10$ $40$ $C_5\times D_{10}$ 5A 2C $bc$
$D_5^3.C_2^2$ 10F $10$ $40$ $C_5\times D_{10}$ 5A 2B $be^{3}$
$D_5^3.C_2^2$ 10G $10$ $40$ $C_5\times C_{20}$ 5D 2A $a^{2}b^{4}d^{4}e$
$D_5^3.C_2^2$ 10H1 $10$ $80$ $C_5\times C_{10}$ 5E1 2C $b^{7}cde^{2}$
$D_5^3.C_2^2$ 10H3 $10$ $80$ $C_5\times C_{10}$ 5E2 2C $bcd$
$D_5^3.C_2^2$ 10I1 $10$ $80$ $C_5\times C_{10}$ 5F1 2B $b^{9}c^{2}d^{2}e^{4}$
$D_5^3.C_2^2$ 10I3 $10$ $80$ $C_5\times C_{10}$ 5F2 2B $b^{7}c^{2}$
$D_5^3.C_2^2$ 10J $10$ $100$ $C_5\times D_4$ 5A 2D $b^{6}c^{2}d^{2}e$
$D_5^3.C_2^2$ 10K1 $10$ $100$ $C_2\times C_{20}$ 5C1 2E $a^{2}bc^{3}de^{3}$
$D_5^3.C_2^2$ 10K3 $10$ $100$ $C_2\times C_{20}$ 5C2 2E $a^{2}bc^{3}d^{3}$
$D_5^3.C_2^2$ 10L1 $10$ $100$ $C_2\times C_{20}$ 5B2 2F $a^{2}b^{5}c^{2}d^{2}e^{2}$
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