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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{2}=b^{9}=c^{2}=d^{10}=[a,d]=[c,d]=1, b^{a}=b^{8}, c^{a}=cd^{5}, c^{b}=cd^{5}, d^{b}=cd^{6} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 5P
$C_{15}.S_4$ 1A $1$ $1$ $C_{15}.S_4$ 1A 1A 1A $1$
$C_{15}.S_4$ 2A $2$ $3$ $C_{15}:D_4$ 1A 2A 2A $d^{5}$
$C_{15}.S_4$ 2B $2$ $18$ $C_2\times C_{10}$ 1A 2B 2B $a$
$C_{15}.S_4$ 3A $3$ $2$ $C_2^2:C_{45}$ 3A 1A 3A $b^{3}$
$C_{15}.S_4$ 4A $4$ $18$ $C_{20}$ 2A 4A 4A $ac$
$C_{15}.S_4$ 5A1 $5$ $1$ $C_{15}.S_4$ 5A2 5A-2 1A $d^{4}$
$C_{15}.S_4$ 5A-1 $5$ $1$ $C_{15}.S_4$ 5A-2 5A2 1A $d^{6}$
$C_{15}.S_4$ 5A2 $5$ $1$ $C_{15}.S_4$ 5A-1 5A1 1A $d^{8}$
$C_{15}.S_4$ 5A-2 $5$ $1$ $C_{15}.S_4$ 5A1 5A-1 1A $d^{2}$
$C_{15}.S_4$ 6A $6$ $6$ $C_2\times C_{30}$ 3A 2A 6A $b^{6}d^{5}$
$C_{15}.S_4$ 9A1 $9$ $8$ $C_{45}$ 9A2 3A 9A4 $b^{5}$
$C_{15}.S_4$ 9A2 $9$ $8$ $C_{45}$ 9A4 3A 9A1 $b$
$C_{15}.S_4$ 9A4 $9$ $8$ $C_{45}$ 9A1 3A 9A2 $b^{2}$
$C_{15}.S_4$ 10A1 $10$ $3$ $C_{15}:D_4$ 5A2 10A3 2A $d^{9}$
$C_{15}.S_4$ 10A-1 $10$ $3$ $C_{15}:D_4$ 5A-2 10A-3 2A $d$
$C_{15}.S_4$ 10A3 $10$ $3$ $C_{15}:D_4$ 5A1 10A-1 2A $d^{7}$
$C_{15}.S_4$ 10A-3 $10$ $3$ $C_{15}:D_4$ 5A-1 10A1 2A $d^{3}$
$C_{15}.S_4$ 10B1 $10$ $18$ $C_2\times C_{10}$ 5A1 10B3 2B $ad^{2}$
$C_{15}.S_4$ 10B-1 $10$ $18$ $C_2\times C_{10}$ 5A-1 10B-3 2B $ad^{8}$
$C_{15}.S_4$ 10B3 $10$ $18$ $C_2\times C_{10}$ 5A-2 10B-1 2B $ad$
$C_{15}.S_4$ 10B-3 $10$ $18$ $C_2\times C_{10}$ 5A2 10B1 2B $ad^{4}$
$C_{15}.S_4$ 15A1 $15$ $2$ $C_2^2:C_{45}$ 15A2 5A-1 3A $b^{6}d^{2}$
$C_{15}.S_4$ 15A-1 $15$ $2$ $C_2^2:C_{45}$ 15A-2 5A1 3A $b^{3}d^{8}$
$C_{15}.S_4$ 15A2 $15$ $2$ $C_2^2:C_{45}$ 15A-1 5A-2 3A $b^{3}d^{4}$
$C_{15}.S_4$ 15A-2 $15$ $2$ $C_2^2:C_{45}$ 15A1 5A2 3A $b^{6}d^{6}$
$C_{15}.S_4$ 20A1 $20$ $18$ $C_{20}$ 10A1 20A3 4A $acd^{2}$
$C_{15}.S_4$ 20A-1 $20$ $18$ $C_{20}$ 10A-1 20A-3 4A $acd^{8}$
$C_{15}.S_4$ 20A3 $20$ $18$ $C_{20}$ 10A3 20A-1 4A $acd$
$C_{15}.S_4$ 20A-3 $20$ $18$ $C_{20}$ 10A-3 20A1 4A $acd^{4}$
$C_{15}.S_4$ 30A1 $30$ $6$ $C_2\times C_{30}$ 15A1 10A-3 6A $b^{3}d$
$C_{15}.S_4$ 30A-1 $30$ $6$ $C_2\times C_{30}$ 15A-1 10A3 6A $b^{3}cd^{4}$
$C_{15}.S_4$ 30A7 $30$ $6$ $C_2\times C_{30}$ 15A2 10A-1 6A $b^{3}cd^{2}$
$C_{15}.S_4$ 30A-7 $30$ $6$ $C_2\times C_{30}$ 15A-2 10A1 6A $b^{3}d^{3}$
$C_{15}.S_4$ 45A1 $45$ $8$ $C_{45}$ 45A2 15A-2 9A1 $bd^{2}$
$C_{15}.S_4$ 45A-1 $45$ $8$ $C_{45}$ 45A-2 15A2 9A1 $bd^{8}$
$C_{15}.S_4$ 45A2 $45$ $8$ $C_{45}$ 45A4 15A1 9A2 $b^{2}d^{4}$
$C_{15}.S_4$ 45A-2 $45$ $8$ $C_{45}$ 45A-4 15A-1 9A2 $b^{2}d$
$C_{15}.S_4$ 45A4 $45$ $8$ $C_{45}$ 45A8 15A2 9A4 $b^{4}d^{8}$
$C_{15}.S_4$ 45A-4 $45$ $8$ $C_{45}$ 45A-8 15A-2 9A4 $b^{4}d^{2}$
$C_{15}.S_4$ 45A8 $45$ $8$ $C_{45}$ 45A11 15A-1 9A1 $bd$
$C_{15}.S_4$ 45A-8 $45$ $8$ $C_{45}$ 45A-11 15A1 9A1 $bd^{4}$
$C_{15}.S_4$ 45A11 $45$ $8$ $C_{45}$ 45A-13 15A-2 9A2 $b^{2}d^{2}$
$C_{15}.S_4$ 45A-11 $45$ $8$ $C_{45}$ 45A13 15A2 9A2 $b^{2}d^{8}$
$C_{15}.S_4$ 45A13 $45$ $8$ $C_{45}$ 45A1 15A-1 9A4 $b^{4}d$
$C_{15}.S_4$ 45A-13 $45$ $8$ $C_{45}$ 45A-1 15A1 9A4 $b^{4}d^{4}$
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