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Elements of the group are displayed as words in the presentation $\langle a, b, c \mid b^{20}=c^{5}=[a,c]=1, a^{2}=b^{10}, b^{a}=b^{9}, c^{b}=c^{4} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 5P
$C_{10}.D_{10}$ 1A $1$ $1$ $C_{10}.D_{10}$ 1A 1A $1$
$C_{10}.D_{10}$ 2A $2$ $1$ $C_{10}.D_{10}$ 1A 2A $b^{10}$
$C_{10}.D_{10}$ 2B $2$ $25$ $C_2\times C_4$ 1A 2B $ab^{17}$
$C_{10}.D_{10}$ 2C $2$ $25$ $C_2\times C_4$ 1A 2C $ab^{19}c$
$C_{10}.D_{10}$ 4A1 $4$ $5$ $C_4\times D_5$ 2A 4A1 $b^{5}$
$C_{10}.D_{10}$ 4A-1 $4$ $5$ $C_4\times D_5$ 2A 4A-1 $b^{15}$
$C_{10}.D_{10}$ 4B1 $4$ $5$ $C_4\times D_5$ 2A 4B1 $a$
$C_{10}.D_{10}$ 4B-1 $4$ $5$ $C_4\times D_5$ 2A 4B-1 $ab^{10}$
$C_{10}.D_{10}$ 5A1 $5$ $2$ $C_5:C_{20}$ 5A2 1A $b^{4}$
$C_{10}.D_{10}$ 5A2 $5$ $2$ $C_5:C_{20}$ 5A1 1A $b^{8}$
$C_{10}.D_{10}$ 5B1 $5$ $2$ $C_5:C_{20}$ 5B2 1A $c^{4}$
$C_{10}.D_{10}$ 5B2 $5$ $2$ $C_5:C_{20}$ 5B1 1A $c^{3}$
$C_{10}.D_{10}$ 5C1 $5$ $4$ $C_5\times C_{10}$ 5C2 1A $b^{4}c^{2}$
$C_{10}.D_{10}$ 5C2 $5$ $4$ $C_5\times C_{10}$ 5C1 1A $b^{8}c^{4}$
$C_{10}.D_{10}$ 5D1 $5$ $4$ $C_5\times C_{10}$ 5D2 1A $b^{4}c^{4}$
$C_{10}.D_{10}$ 5D2 $5$ $4$ $C_5\times C_{10}$ 5D1 1A $b^{8}c^{3}$
$C_{10}.D_{10}$ 10A1 $10$ $2$ $C_5:C_{20}$ 5A1 2A $b^{2}$
$C_{10}.D_{10}$ 10A3 $10$ $2$ $C_5:C_{20}$ 5A2 2A $b^{6}$
$C_{10}.D_{10}$ 10B1 $10$ $2$ $C_5:C_{20}$ 5B1 2A $b^{10}c^{2}$
$C_{10}.D_{10}$ 10B3 $10$ $2$ $C_5:C_{20}$ 5B2 2A $b^{10}c$
$C_{10}.D_{10}$ 10C1 $10$ $4$ $C_5\times C_{10}$ 5C1 2A $b^{2}c$
$C_{10}.D_{10}$ 10C3 $10$ $4$ $C_5\times C_{10}$ 5C2 2A $b^{6}c^{2}$
$C_{10}.D_{10}$ 10D1 $10$ $4$ $C_5\times C_{10}$ 5D1 2A $b^{2}c^{2}$
$C_{10}.D_{10}$ 10D3 $10$ $4$ $C_5\times C_{10}$ 5D2 2A $b^{6}c$
$C_{10}.D_{10}$ 20A1 $20$ $10$ $C_{20}$ 10A1 4A1 $b$
$C_{10}.D_{10}$ 20A-1 $20$ $10$ $C_{20}$ 10A1 4A-1 $b^{11}$
$C_{10}.D_{10}$ 20A3 $20$ $10$ $C_{20}$ 10A3 4A-1 $b^{3}$
$C_{10}.D_{10}$ 20A-3 $20$ $10$ $C_{20}$ 10A3 4A1 $b^{13}$
$C_{10}.D_{10}$ 20B1 $20$ $10$ $C_{20}$ 10B1 4B1 $ac$
$C_{10}.D_{10}$ 20B-1 $20$ $10$ $C_{20}$ 10B1 4B-1 $ab^{2}c$
$C_{10}.D_{10}$ 20B3 $20$ $10$ $C_{20}$ 10B3 4B-1 $ab^{2}c^{2}$
$C_{10}.D_{10}$ 20B-3 $20$ $10$ $C_{20}$ 10B3 4B1 $ac^{2}$
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