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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{3}=b^{3}=c^{3}=d^{45}=[a,b]=[a,c]=[c,d]=1, d^{a}=d^{31}, c^{b}=cd^{15}, d^{b}=cd^{16} \rangle$ .

Group Label Order Size Centralizer Powers Representative
3P 5P
$(C_3\times C_{45}):C_3^2$ 1A $1$ $1$ $(C_3\times C_{45}):C_3^2$ 1A 1A $1$
$(C_3\times C_{45}):C_3^2$ 3A1 $3$ $1$ $(C_3\times C_{45}):C_3^2$ 3A-1 1A $d^{15}$
$(C_3\times C_{45}):C_3^2$ 3A-1 $3$ $1$ $(C_3\times C_{45}):C_3^2$ 3A1 1A $d^{30}$
$(C_3\times C_{45}):C_3^2$ 3B1 $3$ $3$ $C_{45}:C_3^2$ 3B-1 1A $c^{2}$
$(C_3\times C_{45}):C_3^2$ 3B-1 $3$ $3$ $C_{45}:C_3^2$ 3B1 1A $c$
$(C_3\times C_{45}):C_3^2$ 3C1 $3$ $3$ $C_{15}\times \He_3$ 3C-1 1A $a^{2}$
$(C_3\times C_{45}):C_3^2$ 3C-1 $3$ $3$ $C_{15}\times \He_3$ 3C1 1A $a$
$(C_3\times C_{45}):C_3^2$ 3D1 $3$ $3$ $C_{15}\times \He_3$ 3D-1 1A $a^{2}c^{2}$
$(C_3\times C_{45}):C_3^2$ 3D-1 $3$ $3$ $C_{15}\times \He_3$ 3D1 1A $ac$
$(C_3\times C_{45}):C_3^2$ 3E1 $3$ $3$ $C_{15}\times \He_3$ 3E-1 1A $a^{2}c$
$(C_3\times C_{45}):C_3^2$ 3E-1 $3$ $3$ $C_{15}\times \He_3$ 3E1 1A $ac^{2}$
$(C_3\times C_{45}):C_3^2$ 3F1 $3$ $9$ $C_3^2\times C_{15}$ 3F-1 1A $b^{2}$
$(C_3\times C_{45}):C_3^2$ 3F-1 $3$ $9$ $C_3^2\times C_{15}$ 3F1 1A $b$
$(C_3\times C_{45}):C_3^2$ 3G1 $3$ $9$ $C_3^2\times C_{15}$ 3G-1 1A $b^{2}cd^{35}$
$(C_3\times C_{45}):C_3^2$ 3G-1 $3$ $9$ $C_3^2\times C_{15}$ 3G1 1A $bd^{10}$
$(C_3\times C_{45}):C_3^2$ 3H1 $3$ $9$ $C_3^2\times C_{15}$ 3H-1 1A $b^{2}c^{2}d^{25}$
$(C_3\times C_{45}):C_3^2$ 3H-1 $3$ $9$ $C_3^2\times C_{15}$ 3H1 1A $bd^{20}$
$(C_3\times C_{45}):C_3^2$ 3I1 $3$ $9$ $C_3^2\times C_{15}$ 3I-1 1A $a^{2}b^{2}$
$(C_3\times C_{45}):C_3^2$ 3I-1 $3$ $9$ $C_3^2\times C_{15}$ 3I1 1A $ab$
$(C_3\times C_{45}):C_3^2$ 3J1 $3$ $9$ $C_3^2\times C_{15}$ 3J-1 1A $a^{2}b^{2}cd^{20}$
$(C_3\times C_{45}):C_3^2$ 3J-1 $3$ $9$ $C_3^2\times C_{15}$ 3J1 1A $abd^{10}$
$(C_3\times C_{45}):C_3^2$ 3K1 $3$ $9$ $C_3^2\times C_{15}$ 3K-1 1A $a^{2}b^{2}c^{2}d^{40}$
$(C_3\times C_{45}):C_3^2$ 3K-1 $3$ $9$ $C_3^2\times C_{15}$ 3K1 1A $abd^{20}$
$(C_3\times C_{45}):C_3^2$ 3L1 $3$ $9$ $C_3^2\times C_{15}$ 3L-1 1A $a^{2}b$
$(C_3\times C_{45}):C_3^2$ 3L-1 $3$ $9$ $C_3^2\times C_{15}$ 3L1 1A $ab^{2}$
$(C_3\times C_{45}):C_3^2$ 3M1 $3$ $9$ $C_3^2\times C_{15}$ 3M-1 1A $a^{2}bc^{2}d^{5}$
$(C_3\times C_{45}):C_3^2$ 3M-1 $3$ $9$ $C_3^2\times C_{15}$ 3M1 1A $ab^{2}d^{10}$
$(C_3\times C_{45}):C_3^2$ 3N1 $3$ $9$ $C_3^2\times C_{15}$ 3N-1 1A $ab^{2}cd^{5}$
$(C_3\times C_{45}):C_3^2$ 3N-1 $3$ $9$ $C_3^2\times C_{15}$ 3N1 1A $a^{2}bd^{10}$
$(C_3\times C_{45}):C_3^2$ 5A1 $5$ $1$ $(C_3\times C_{45}):C_3^2$ 5A2 5A-2 $d^{27}$
$(C_3\times C_{45}):C_3^2$ 5A-1 $5$ $1$ $(C_3\times C_{45}):C_3^2$ 5A-2 5A2 $d^{18}$
$(C_3\times C_{45}):C_3^2$ 5A2 $5$ $1$ $(C_3\times C_{45}):C_3^2$ 5A-1 5A1 $d^{9}$
$(C_3\times C_{45}):C_3^2$ 5A-2 $5$ $1$ $(C_3\times C_{45}):C_3^2$ 5A1 5A-1 $d^{36}$
$(C_3\times C_{45}):C_3^2$ 9A1 $9$ $9$ $C_3\times C_{45}$ 9A-1 3A1 $d^{5}$
$(C_3\times C_{45}):C_3^2$ 9A-1 $9$ $9$ $C_3\times C_{45}$ 9A1 3A-1 $d^{40}$
$(C_3\times C_{45}):C_3^2$ 9B1 $9$ $9$ $C_3\times C_{45}$ 9B-1 3A1 $a^{2}d^{35}$
$(C_3\times C_{45}):C_3^2$ 9B-1 $9$ $9$ $C_3\times C_{45}$ 9B1 3A-1 $ad^{40}$
$(C_3\times C_{45}):C_3^2$ 9C1 $9$ $9$ $C_3\times C_{45}$ 9C-1 3A-1 $a^{2}d^{25}$
$(C_3\times C_{45}):C_3^2$ 9C-1 $9$ $9$ $C_3\times C_{45}$ 9C1 3A1 $ad^{35}$
$(C_3\times C_{45}):C_3^2$ 15A1 $15$ $1$ $(C_3\times C_{45}):C_3^2$ 15A2 5A2 $d^{3}$
$(C_3\times C_{45}):C_3^2$ 15A-1 $15$ $1$ $(C_3\times C_{45}):C_3^2$ 15A-2 5A-2 $d^{42}$
$(C_3\times C_{45}):C_3^2$ 15A2 $15$ $1$ $(C_3\times C_{45}):C_3^2$ 15A4 5A-1 $d^{6}$
$(C_3\times C_{45}):C_3^2$ 15A-2 $15$ $1$ $(C_3\times C_{45}):C_3^2$ 15A-4 5A1 $d^{39}$
$(C_3\times C_{45}):C_3^2$ 15A4 $15$ $1$ $(C_3\times C_{45}):C_3^2$ 15A-7 5A-2 $d^{12}$
$(C_3\times C_{45}):C_3^2$ 15A-4 $15$ $1$ $(C_3\times C_{45}):C_3^2$ 15A7 5A2 $d^{33}$
$(C_3\times C_{45}):C_3^2$ 15A7 $15$ $1$ $(C_3\times C_{45}):C_3^2$ 15A-1 5A-1 $d^{21}$
$(C_3\times C_{45}):C_3^2$ 15A-7 $15$ $1$ $(C_3\times C_{45}):C_3^2$ 15A1 5A1 $d^{24}$
$(C_3\times C_{45}):C_3^2$ 15B1 $15$ $3$ $C_{45}:C_3^2$ 15B2 5A1 $cd^{9}$
$(C_3\times C_{45}):C_3^2$ 15B-1 $15$ $3$ $C_{45}:C_3^2$ 15B-2 5A-1 $c^{2}d^{6}$
$(C_3\times C_{45}):C_3^2$ 15B2 $15$ $3$ $C_{45}:C_3^2$ 15B4 5A2 $c^{2}d^{3}$
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