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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{2}=b^{3}=c^{15}=d^{5}=[a,d]=[c,d]=1, b^{a}=b^{2}, c^{a}=c^{4}d^{4}, c^{b}=c^{13}d^{4}, d^{b}=c^{3}d \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 5P
$C_3\times C_5^2:S_3$ 1A $1$ $1$ $C_3\times C_5^2:S_3$ 1A 1A 1A $1$
$C_3\times C_5^2:S_3$ 2A $2$ $15$ $C_{30}$ 1A 2A 2A $ac^{6}d^{3}$
$C_3\times C_5^2:S_3$ 3A1 $3$ $1$ $C_3\times C_5^2:S_3$ 3A-1 1A 3A-1 $c^{5}$
$C_3\times C_5^2:S_3$ 3A-1 $3$ $1$ $C_3\times C_5^2:S_3$ 3A1 1A 3A1 $c^{10}$
$C_3\times C_5^2:S_3$ 3B $3$ $50$ $C_3^2$ 3B 1A 3B $b^{2}$
$C_3\times C_5^2:S_3$ 3C1 $3$ $50$ $C_3^2$ 3C-1 1A 3C-1 $bc^{14}d^{3}$
$C_3\times C_5^2:S_3$ 3C-1 $3$ $50$ $C_3^2$ 3C1 1A 3C1 $b^{2}c^{10}d^{2}$
$C_3\times C_5^2:S_3$ 5A1 $5$ $3$ $D_5\times C_{15}$ 5A2 5A-2 1A $d^{2}$
$C_3\times C_5^2:S_3$ 5A-1 $5$ $3$ $D_5\times C_{15}$ 5A-2 5A2 1A $d^{3}$
$C_3\times C_5^2:S_3$ 5A2 $5$ $3$ $D_5\times C_{15}$ 5A-1 5A1 1A $d^{4}$
$C_3\times C_5^2:S_3$ 5A-2 $5$ $3$ $D_5\times C_{15}$ 5A1 5A-1 1A $d$
$C_3\times C_5^2:S_3$ 5B1 $5$ $6$ $C_5\times C_{15}$ 5B2 5B2 1A $c^{3}$
$C_3\times C_5^2:S_3$ 5B2 $5$ $6$ $C_5\times C_{15}$ 5B1 5B1 1A $c^{6}$
$C_3\times C_5^2:S_3$ 6A1 $6$ $15$ $C_{30}$ 3A-1 2A 6A-1 $ac^{11}d^{3}$
$C_3\times C_5^2:S_3$ 6A-1 $6$ $15$ $C_{30}$ 3A1 2A 6A1 $acd^{3}$
$C_3\times C_5^2:S_3$ 10A1 $10$ $15$ $C_{30}$ 5A1 10A3 2A $ac^{6}d^{4}$
$C_3\times C_5^2:S_3$ 10A-1 $10$ $15$ $C_{30}$ 5A-1 10A-3 2A $ac^{6}d^{2}$
$C_3\times C_5^2:S_3$ 10A3 $10$ $15$ $C_{30}$ 5A-2 10A-1 2A $ac^{6}d$
$C_3\times C_5^2:S_3$ 10A-3 $10$ $15$ $C_{30}$ 5A2 10A1 2A $ac^{6}$
$C_3\times C_5^2:S_3$ 15A1 $15$ $3$ $D_5\times C_{15}$ 15A2 5A1 3A-1 $c^{5}d^{4}$
$C_3\times C_5^2:S_3$ 15A-1 $15$ $3$ $D_5\times C_{15}$ 15A-2 5A-1 3A1 $c^{10}d$
$C_3\times C_5^2:S_3$ 15A2 $15$ $3$ $D_5\times C_{15}$ 15A4 5A2 3A1 $c^{10}d^{3}$
$C_3\times C_5^2:S_3$ 15A-2 $15$ $3$ $D_5\times C_{15}$ 15A-4 5A-2 3A-1 $c^{5}d^{2}$
$C_3\times C_5^2:S_3$ 15A4 $15$ $3$ $D_5\times C_{15}$ 15A-7 5A-1 3A-1 $c^{5}d$
$C_3\times C_5^2:S_3$ 15A-4 $15$ $3$ $D_5\times C_{15}$ 15A7 5A1 3A1 $c^{10}d^{4}$
$C_3\times C_5^2:S_3$ 15A7 $15$ $3$ $D_5\times C_{15}$ 15A-1 5A2 3A-1 $c^{5}d^{3}$
$C_3\times C_5^2:S_3$ 15A-7 $15$ $3$ $D_5\times C_{15}$ 15A1 5A-2 3A1 $c^{10}d^{2}$
$C_3\times C_5^2:S_3$ 15B1 $15$ $6$ $C_5\times C_{15}$ 15B2 5B1 3A1 $c$
$C_3\times C_5^2:S_3$ 15B-1 $15$ $6$ $C_5\times C_{15}$ 15B-2 5B1 3A-1 $c^{2}d$
$C_3\times C_5^2:S_3$ 15B2 $15$ $6$ $C_5\times C_{15}$ 15B1 5B2 3A-1 $c^{2}$
$C_3\times C_5^2:S_3$ 15B-2 $15$ $6$ $C_5\times C_{15}$ 15B-1 5B2 3A1 $c^{7}$
$C_3\times C_5^2:S_3$ 30A1 $30$ $15$ $C_{30}$ 15A1 10A1 6A1 $ac$
$C_3\times C_5^2:S_3$ 30A-1 $30$ $15$ $C_{30}$ 15A-1 10A-1 6A-1 $ac^{2}d^{4}$
$C_3\times C_5^2:S_3$ 30A7 $30$ $15$ $C_{30}$ 15A7 10A-3 6A1 $acd^{2}$
$C_3\times C_5^2:S_3$ 30A-7 $30$ $15$ $C_{30}$ 15A-7 10A3 6A-1 $ac^{2}d^{2}$
$C_3\times C_5^2:S_3$ 30A11 $30$ $15$ $C_{30}$ 15A-4 10A1 6A-1 $ac^{2}d^{3}$
$C_3\times C_5^2:S_3$ 30A-11 $30$ $15$ $C_{30}$ 15A4 10A-1 6A1 $acd$
$C_3\times C_5^2:S_3$ 30A13 $30$ $15$ $C_{30}$ 15A-2 10A3 6A1 $acd^{4}$
$C_3\times C_5^2:S_3$ 30A-13 $30$ $15$ $C_{30}$ 15A2 10A-3 6A-1 $ac^{2}$
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