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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid b^{2}=c^{8}=d^{10}=[b,d]=1, a^{2}=c^{4}, b^{a}=bc^{4}, c^{a}=c^{3}, d^{a}=c^{4}d, c^{b}=c^{3}, d^{c}=d^{9} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 5P
$C_{20}.C_2^4$ 1A $1$ $1$ $C_{20}.C_2^4$ 1A 1A $1$
$C_{20}.C_2^4$ 2A $2$ $1$ $C_{20}.C_2^4$ 1A 2A $c^{4}$
$C_{20}.C_2^4$ 2B $2$ $2$ $C_{10}:\SD_{16}$ 1A 2B $d^{5}$
$C_{20}.C_2^4$ 2C $2$ $2$ $C_{10}:\SD_{16}$ 1A 2C $ab$
$C_{20}.C_2^4$ 2D $2$ $2$ $C_{10}:\SD_{16}$ 1A 2D $abc^{2}d^{5}$
$C_{20}.C_2^4$ 2E $2$ $4$ $D_4\times C_{10}$ 1A 2E $b$
$C_{20}.C_2^4$ 2F $2$ $4$ $D_4\times C_{10}$ 1A 2F $bd^{5}$
$C_{20}.C_2^4$ 2G $2$ $4$ $D_4\times C_{10}$ 1A 2G $ad^{5}$
$C_{20}.C_2^4$ 2H $2$ $20$ $C_2\times Q_8$ 1A 2H $ac^{3}$
$C_{20}.C_2^4$ 4A $4$ $2$ $C_{20}.C_2^3$ 2A 4A $c^{2}$
$C_{20}.C_2^4$ 4B $4$ $2$ $C_{20}.D_4$ 2A 4B $c^{2}d^{5}$
$C_{20}.C_2^4$ 4C $4$ $2$ $C_{20}.D_4$ 2A 4C $abd^{5}$
$C_{20}.C_2^4$ 4D $4$ $2$ $C_{20}.D_4$ 2A 4D $abc^{2}$
$C_{20}.C_2^4$ 4E $4$ $4$ $D_4:C_{10}$ 2A 4E $a$
$C_{20}.C_2^4$ 4F $4$ $20$ $D_4:C_2$ 2A 4F $ac^{5}d^{5}$
$C_{20}.C_2^4$ 4G $4$ $20$ $D_4:C_2$ 2A 4G $bc^{3}d^{6}$
$C_{20}.C_2^4$ 4H $4$ $20$ $D_4:C_2$ 2A 4H $bc^{5}d^{9}$
$C_{20}.C_2^4$ 5A1 $5$ $2$ $C_{10}.C_2^4$ 5A2 1A $d^{2}$
$C_{20}.C_2^4$ 5A2 $5$ $2$ $C_{10}.C_2^4$ 5A1 1A $d^{4}$
$C_{20}.C_2^4$ 8A1 $8$ $10$ $\OD_{16}:C_2$ 4A 8A-1 $c$
$C_{20}.C_2^4$ 8A-1 $8$ $10$ $\OD_{16}:C_2$ 4A 8A1 $c^{5}$
$C_{20}.C_2^4$ 8B $8$ $20$ $C_2\times C_8$ 4A 8B $abcd^{9}$
$C_{20}.C_2^4$ 8C $8$ $20$ $C_2\times C_8$ 4A 8C $abc^{5}d^{6}$
$C_{20}.C_2^4$ 8D $8$ $20$ $C_2\times C_8$ 4A 8D $c^{5}d^{9}$
$C_{20}.C_2^4$ 10A1 $10$ $2$ $C_{10}.C_2^4$ 5A1 2A $c^{4}d^{4}$
$C_{20}.C_2^4$ 10A3 $10$ $2$ $C_{10}.C_2^4$ 5A2 2A $c^{4}d^{2}$
$C_{20}.C_2^4$ 10B1 $10$ $4$ $D_4\times C_{10}$ 5A1 2B $d$
$C_{20}.C_2^4$ 10B3 $10$ $4$ $D_4\times C_{10}$ 5A2 2B $d^{3}$
$C_{20}.C_2^4$ 10C1 $10$ $4$ $D_4\times C_{10}$ 5A2 2C $abd^{2}$
$C_{20}.C_2^4$ 10C3 $10$ $4$ $D_4\times C_{10}$ 5A1 2C $abd^{4}$
$C_{20}.C_2^4$ 10D1 $10$ $4$ $D_4\times C_{10}$ 5A1 2D $abc^{2}d$
$C_{20}.C_2^4$ 10D3 $10$ $4$ $D_4\times C_{10}$ 5A2 2D $abc^{2}d^{3}$
$C_{20}.C_2^4$ 10E1 $10$ $4$ $D_4\times C_{10}$ 5A2 2E $bd^{2}$
$C_{20}.C_2^4$ 10E-1 $10$ $4$ $D_4\times C_{10}$ 5A2 2E $bd^{8}$
$C_{20}.C_2^4$ 10E3 $10$ $4$ $D_4\times C_{10}$ 5A1 2E $bd^{6}$
$C_{20}.C_2^4$ 10E-3 $10$ $4$ $D_4\times C_{10}$ 5A1 2E $bd^{4}$
$C_{20}.C_2^4$ 10F1 $10$ $4$ $D_4\times C_{10}$ 5A1 2F $bd$
$C_{20}.C_2^4$ 10F-1 $10$ $4$ $D_4\times C_{10}$ 5A1 2F $bc^{2}d$
$C_{20}.C_2^4$ 10F3 $10$ $4$ $D_4\times C_{10}$ 5A2 2F $bd^{3}$
$C_{20}.C_2^4$ 10F-3 $10$ $4$ $D_4\times C_{10}$ 5A2 2F $bd^{7}$
$C_{20}.C_2^4$ 10G1 $10$ $4$ $D_4\times C_{10}$ 5A1 2G $ad$
$C_{20}.C_2^4$ 10G-1 $10$ $4$ $D_4\times C_{10}$ 5A1 2G $ac^{2}d$
$C_{20}.C_2^4$ 10G3 $10$ $4$ $D_4\times C_{10}$ 5A2 2G $ad^{3}$
$C_{20}.C_2^4$ 10G-3 $10$ $4$ $D_4\times C_{10}$ 5A2 2G $ad^{7}$
$C_{20}.C_2^4$ 20A1 $20$ $4$ $D_4:C_{10}$ 10A1 4A $c^{2}d^{2}$
$C_{20}.C_2^4$ 20A3 $20$ $4$ $D_4:C_{10}$ 10A3 4A $c^{2}d^{4}$
$C_{20}.C_2^4$ 20B1 $20$ $4$ $D_4:C_{10}$ 10A3 4B $c^{2}d$
$C_{20}.C_2^4$ 20B3 $20$ $4$ $D_4:C_{10}$ 10A1 4B $c^{2}d^{3}$
$C_{20}.C_2^4$ 20C1 $20$ $4$ $D_4:C_{10}$ 10A3 4C $abd$
$C_{20}.C_2^4$ 20C3 $20$ $4$ $D_4:C_{10}$ 10A1 4C $abd^{3}$
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