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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{4}=b^{4}=c^{5}=d^{10}=[a,c]=[a,d]=[c,d]=1, b^{a}=b^{3}, c^{b}=c^{4}, d^{b}=d^{9} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 5P
$C_{10}^2.D_4$ 1A $1$ $1$ $C_{10}^2.D_4$ 1A 1A $1$
$C_{10}^2.D_4$ 2A $2$ $1$ $C_{10}^2.D_4$ 1A 2A $d^{5}$
$C_{10}^2.D_4$ 2B $2$ $1$ $C_{10}^2.D_4$ 1A 2B $b^{2}$
$C_{10}^2.D_4$ 2C $2$ $1$ $C_{10}^2.D_4$ 1A 2C $b^{2}d^{5}$
$C_{10}^2.D_4$ 2D $2$ $1$ $C_{10}^2.D_4$ 1A 2D $a^{2}d^{5}$
$C_{10}^2.D_4$ 2E $2$ $1$ $C_{10}^2.D_4$ 1A 2E $a^{2}b^{2}$
$C_{10}^2.D_4$ 2F $2$ $1$ $C_{10}^2.D_4$ 1A 2F $a^{2}b^{2}d^{5}$
$C_{10}^2.D_4$ 2G $2$ $1$ $C_{10}^2.D_4$ 1A 2G $a^{2}$
$C_{10}^2.D_4$ 4A1 $4$ $2$ $C_2\times C_{10}\times C_{20}$ 2G 4A1 $a$
$C_{10}^2.D_4$ 4A-1 $4$ $2$ $C_2\times C_{10}\times C_{20}$ 2G 4A-1 $a^{3}$
$C_{10}^2.D_4$ 4B1 $4$ $2$ $C_2\times C_{10}\times C_{20}$ 2G 4B1 $ad^{5}$
$C_{10}^2.D_4$ 4B-1 $4$ $2$ $C_2\times C_{10}\times C_{20}$ 2G 4B-1 $a^{3}d^{5}$
$C_{10}^2.D_4$ 4C $4$ $50$ $C_2^2\times C_4$ 2B 4C $a^{2}bd^{6}$
$C_{10}^2.D_4$ 4D $4$ $50$ $C_2^2\times C_4$ 2B 4D $a^{2}bc^{4}d^{9}$
$C_{10}^2.D_4$ 4E $4$ $50$ $C_2^2\times C_4$ 2B 4E $b^{3}c^{4}d^{9}$
$C_{10}^2.D_4$ 4F $4$ $50$ $C_2^2\times C_4$ 2B 4F $bc$
$C_{10}^2.D_4$ 4G1 $4$ $50$ $C_2^2\times C_4$ 2G 4G1 $ab^{3}$
$C_{10}^2.D_4$ 4G-1 $4$ $50$ $C_2^2\times C_4$ 2G 4G-1 $a^{3}b^{3}$
$C_{10}^2.D_4$ 4H1 $4$ $50$ $C_2^2\times C_4$ 2G 4H1 $a^{3}bd^{9}$
$C_{10}^2.D_4$ 4H-1 $4$ $50$ $C_2^2\times C_4$ 2G 4H-1 $abd^{9}$
$C_{10}^2.D_4$ 5A1 $5$ $2$ $C_2\times C_{10}\times C_{20}$ 5A2 1A $d^{2}$
$C_{10}^2.D_4$ 5A2 $5$ $2$ $C_2\times C_{10}\times C_{20}$ 5A1 1A $d^{4}$
$C_{10}^2.D_4$ 5B1 $5$ $2$ $C_2\times C_{10}\times C_{20}$ 5B2 1A $c^{2}d^{2}$
$C_{10}^2.D_4$ 5B2 $5$ $2$ $C_2\times C_{10}\times C_{20}$ 5B1 1A $c^{4}d^{4}$
$C_{10}^2.D_4$ 5C1 $5$ $2$ $C_2\times C_{10}\times C_{20}$ 5C2 1A $c^{2}$
$C_{10}^2.D_4$ 5C2 $5$ $2$ $C_2\times C_{10}\times C_{20}$ 5C1 1A $c^{4}$
$C_{10}^2.D_4$ 5D1 $5$ $2$ $C_2\times C_{10}\times C_{20}$ 5D2 1A $c^{4}d^{2}$
$C_{10}^2.D_4$ 5D2 $5$ $2$ $C_2\times C_{10}\times C_{20}$ 5D1 1A $c^{3}d^{4}$
$C_{10}^2.D_4$ 5E1 $5$ $2$ $C_2\times C_{10}\times C_{20}$ 5E2 1A $cd^{2}$
$C_{10}^2.D_4$ 5E2 $5$ $2$ $C_2\times C_{10}\times C_{20}$ 5E1 1A $c^{2}d^{4}$
$C_{10}^2.D_4$ 5F1 $5$ $2$ $C_2\times C_{10}\times C_{20}$ 5F2 1A $c^{3}d^{2}$
$C_{10}^2.D_4$ 5F2 $5$ $2$ $C_2\times C_{10}\times C_{20}$ 5F1 1A $cd^{4}$
$C_{10}^2.D_4$ 10A1 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5A1 2A $d$
$C_{10}^2.D_4$ 10A3 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5A2 2A $d^{3}$
$C_{10}^2.D_4$ 10B1 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5B1 2A $cd$
$C_{10}^2.D_4$ 10B3 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5B2 2A $c^{3}d^{3}$
$C_{10}^2.D_4$ 10C1 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5A2 2B $b^{2}d^{2}$
$C_{10}^2.D_4$ 10C3 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5A1 2B $b^{2}d^{4}$
$C_{10}^2.D_4$ 10D1 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5A1 2C $b^{2}d$
$C_{10}^2.D_4$ 10D3 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5A2 2C $b^{2}d^{3}$
$C_{10}^2.D_4$ 10E1 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5C1 2B $b^{2}c$
$C_{10}^2.D_4$ 10E3 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5C2 2B $b^{2}c^{2}$
$C_{10}^2.D_4$ 10F1 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5A1 2D $a^{2}d$
$C_{10}^2.D_4$ 10F3 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5A2 2D $a^{2}d^{3}$
$C_{10}^2.D_4$ 10G1 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5D1 2A $c^{2}d$
$C_{10}^2.D_4$ 10G3 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5D2 2A $cd^{3}$
$C_{10}^2.D_4$ 10H1 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5E1 2A $c^{3}d$
$C_{10}^2.D_4$ 10H3 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5E2 2A $cd^{7}$
$C_{10}^2.D_4$ 10I1 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5F1 2A $c^{4}d$
$C_{10}^2.D_4$ 10I3 $10$ $2$ $C_2\times C_{10}\times C_{20}$ 5F2 2A $c^{2}d^{3}$
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