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Elements of the group are displayed as words in the presentation of the abelian group $\langle a, b \mid a^{3}=b^{15}=1 \rangle$ .

Group Label Order Size Centralizer Powers Representative
3P 5P
$C_3\times C_{15}$ 1A $1$ $1$ $C_3\times C_{15}$ 1A 1A $1$
$C_3\times C_{15}$ 3A1 $3$ $1$ $C_3\times C_{15}$ 3A-1 1A $b^{5}$
$C_3\times C_{15}$ 3A-1 $3$ $1$ $C_3\times C_{15}$ 3A1 1A $b^{10}$
$C_3\times C_{15}$ 3B1 $3$ $1$ $C_3\times C_{15}$ 3B-1 1A $a^{2}$
$C_3\times C_{15}$ 3B-1 $3$ $1$ $C_3\times C_{15}$ 3B1 1A $a$
$C_3\times C_{15}$ 3C1 $3$ $1$ $C_3\times C_{15}$ 3C-1 1A $a^{2}b^{5}$
$C_3\times C_{15}$ 3C-1 $3$ $1$ $C_3\times C_{15}$ 3C1 1A $ab^{10}$
$C_3\times C_{15}$ 3D1 $3$ $1$ $C_3\times C_{15}$ 3D-1 1A $a^{2}b^{10}$
$C_3\times C_{15}$ 3D-1 $3$ $1$ $C_3\times C_{15}$ 3D1 1A $ab^{5}$
$C_3\times C_{15}$ 5A1 $5$ $1$ $C_3\times C_{15}$ 5A2 5A-2 $b^{3}$
$C_3\times C_{15}$ 5A-1 $5$ $1$ $C_3\times C_{15}$ 5A-2 5A2 $b^{12}$
$C_3\times C_{15}$ 5A2 $5$ $1$ $C_3\times C_{15}$ 5A-1 5A1 $b^{6}$
$C_3\times C_{15}$ 5A-2 $5$ $1$ $C_3\times C_{15}$ 5A1 5A-1 $b^{9}$
$C_3\times C_{15}$ 15A1 $15$ $1$ $C_3\times C_{15}$ 15A2 5A1 $b$
$C_3\times C_{15}$ 15A-1 $15$ $1$ $C_3\times C_{15}$ 15A-2 5A-1 $b^{14}$
$C_3\times C_{15}$ 15A2 $15$ $1$ $C_3\times C_{15}$ 15A4 5A2 $b^{2}$
$C_3\times C_{15}$ 15A-2 $15$ $1$ $C_3\times C_{15}$ 15A-4 5A-2 $b^{13}$
$C_3\times C_{15}$ 15A4 $15$ $1$ $C_3\times C_{15}$ 15A-7 5A-1 $b^{4}$
$C_3\times C_{15}$ 15A-4 $15$ $1$ $C_3\times C_{15}$ 15A7 5A1 $b^{11}$
$C_3\times C_{15}$ 15A7 $15$ $1$ $C_3\times C_{15}$ 15A-1 5A2 $b^{7}$
$C_3\times C_{15}$ 15A-7 $15$ $1$ $C_3\times C_{15}$ 15A1 5A-2 $b^{8}$
$C_3\times C_{15}$ 15B1 $15$ $1$ $C_3\times C_{15}$ 15B2 5A-2 $ab^{3}$
$C_3\times C_{15}$ 15B-1 $15$ $1$ $C_3\times C_{15}$ 15B-2 5A2 $a^{2}b^{12}$
$C_3\times C_{15}$ 15B2 $15$ $1$ $C_3\times C_{15}$ 15B4 5A1 $a^{2}b^{6}$
$C_3\times C_{15}$ 15B-2 $15$ $1$ $C_3\times C_{15}$ 15B-4 5A-1 $ab^{9}$
$C_3\times C_{15}$ 15B4 $15$ $1$ $C_3\times C_{15}$ 15B-7 5A2 $ab^{12}$
$C_3\times C_{15}$ 15B-4 $15$ $1$ $C_3\times C_{15}$ 15B7 5A-2 $a^{2}b^{3}$
$C_3\times C_{15}$ 15B7 $15$ $1$ $C_3\times C_{15}$ 15B-1 5A1 $ab^{6}$
$C_3\times C_{15}$ 15B-7 $15$ $1$ $C_3\times C_{15}$ 15B1 5A-1 $a^{2}b^{9}$
$C_3\times C_{15}$ 15C1 $15$ $1$ $C_3\times C_{15}$ 15C2 5A1 $ab$
$C_3\times C_{15}$ 15C-1 $15$ $1$ $C_3\times C_{15}$ 15C-2 5A-1 $a^{2}b^{14}$
$C_3\times C_{15}$ 15C2 $15$ $1$ $C_3\times C_{15}$ 15C4 5A2 $a^{2}b^{2}$
$C_3\times C_{15}$ 15C-2 $15$ $1$ $C_3\times C_{15}$ 15C-4 5A-2 $ab^{13}$
$C_3\times C_{15}$ 15C4 $15$ $1$ $C_3\times C_{15}$ 15C-7 5A-1 $ab^{4}$
$C_3\times C_{15}$ 15C-4 $15$ $1$ $C_3\times C_{15}$ 15C7 5A1 $a^{2}b^{11}$
$C_3\times C_{15}$ 15C7 $15$ $1$ $C_3\times C_{15}$ 15C-1 5A2 $ab^{7}$
$C_3\times C_{15}$ 15C-7 $15$ $1$ $C_3\times C_{15}$ 15C1 5A-2 $a^{2}b^{8}$
$C_3\times C_{15}$ 15D1 $15$ $1$ $C_3\times C_{15}$ 15D2 5A2 $ab^{2}$
$C_3\times C_{15}$ 15D-1 $15$ $1$ $C_3\times C_{15}$ 15D-2 5A-2 $a^{2}b^{13}$
$C_3\times C_{15}$ 15D2 $15$ $1$ $C_3\times C_{15}$ 15D4 5A-1 $a^{2}b^{4}$
$C_3\times C_{15}$ 15D-2 $15$ $1$ $C_3\times C_{15}$ 15D-4 5A1 $ab^{11}$
$C_3\times C_{15}$ 15D4 $15$ $1$ $C_3\times C_{15}$ 15D-7 5A-2 $ab^{8}$
$C_3\times C_{15}$ 15D-4 $15$ $1$ $C_3\times C_{15}$ 15D7 5A2 $a^{2}b^{7}$
$C_3\times C_{15}$ 15D7 $15$ $1$ $C_3\times C_{15}$ 15D-1 5A-1 $ab^{14}$
$C_3\times C_{15}$ 15D-7 $15$ $1$ $C_3\times C_{15}$ 15D1 5A1 $a^{2}b$
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