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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{40}=b^{11}=c^{11}=d^{55}=[b,c]=[b,d]=[c,d]=1, b^{a}=b^{8}c^{5}d^{35}, c^{a}=c^{6}, d^{a}=b^{5}c^{2}d^{36} \rangle$ .
| Group | Label | Order | Size | Centralizer | Powers | Representative | ||
|---|---|---|---|---|---|---|---|---|
| 2P | 5P | 11P | ||||||
| $C_5\times C_{11}^3:C_{40}$ | 1A | $1$ | $1$ | $C_5\times C_{11}^3:C_{40}$ | 1A | 1A | 1A | $1$ |
| $C_5\times C_{11}^3:C_{40}$ | 2A | $2$ | $1$ | $C_5\times C_{11}^3:C_{40}$ | 1A | 2A | 2A | $a^{20}$ |
| $C_5\times C_{11}^3:C_{40}$ | 4A1 | $4$ | $121$ | $C_{55}:C_{40}$ | 2A | 4A1 | 4A-1 | $a^{10}b^{3}c^{9}d^{45}$ |
| $C_5\times C_{11}^3:C_{40}$ | 4A-1 | $4$ | $121$ | $C_{55}:C_{40}$ | 2A | 4A-1 | 4A1 | $a^{30}b^{3}c^{9}d^{45}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5A1 | $5$ | $1$ | $C_5\times C_{11}^3:C_{40}$ | 5A2 | 1A | 5A1 | $d^{22}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5A-1 | $5$ | $1$ | $C_5\times C_{11}^3:C_{40}$ | 5A-2 | 1A | 5A-1 | $d^{33}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5A2 | $5$ | $1$ | $C_5\times C_{11}^3:C_{40}$ | 5A-1 | 1A | 5A2 | $d^{44}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5A-2 | $5$ | $1$ | $C_5\times C_{11}^3:C_{40}$ | 5A1 | 1A | 5A-2 | $d^{11}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5B1 | $5$ | $1331$ | $C_5\times C_{40}$ | 5B2 | 1A | 5B1 | $a^{24}b^{10}c^{8}d^{18}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5B-1 | $5$ | $1331$ | $C_5\times C_{40}$ | 5B-2 | 1A | 5B-1 | $a^{16}b^{5}c^{4}d^{42}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5B2 | $5$ | $1331$ | $C_5\times C_{40}$ | 5B-1 | 1A | 5B2 | $a^{8}bc^{3}d^{26}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5B-2 | $5$ | $1331$ | $C_5\times C_{40}$ | 5B1 | 1A | 5B-2 | $a^{32}b^{8}c^{2}d^{54}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5C1 | $5$ | $1331$ | $C_5\times C_{40}$ | 5C2 | 1A | 5C1 | $a^{8}b^{3}c^{8}d^{34}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5C-1 | $5$ | $1331$ | $C_5\times C_{40}$ | 5C-2 | 1A | 5C-1 | $a^{32}b^{2}c^{9}d^{41}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5C2 | $5$ | $1331$ | $C_5\times C_{40}$ | 5C-1 | 1A | 5C2 | $a^{16}b^{4}c^{7}d^{38}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5C-2 | $5$ | $1331$ | $C_5\times C_{40}$ | 5C1 | 1A | 5C-2 | $a^{24}b^{8}c^{3}d^{32}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5D1 | $5$ | $1331$ | $C_5\times C_{40}$ | 5D2 | 1A | 5D1 | $a^{24}b^{10}c^{8}d^{26}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5D-1 | $5$ | $1331$ | $C_5\times C_{40}$ | 5D-2 | 1A | 5D-1 | $a^{16}b^{5}c^{4}d^{24}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5D2 | $5$ | $1331$ | $C_5\times C_{40}$ | 5D-1 | 1A | 5D2 | $a^{8}bc^{3}d^{7}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5D-2 | $5$ | $1331$ | $C_5\times C_{40}$ | 5D1 | 1A | 5D-2 | $a^{32}b^{8}c^{2}d^{23}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5E1 | $5$ | $1331$ | $C_5\times C_{40}$ | 5E2 | 1A | 5E1 | $a^{32}b^{9}c^{4}d^{30}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5E-1 | $5$ | $1331$ | $C_5\times C_{40}$ | 5E-2 | 1A | 5E-1 | $a^{8}b^{8}c^{6}d^{45}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5E2 | $5$ | $1331$ | $C_5\times C_{40}$ | 5E-1 | 1A | 5E2 | $a^{24}b^{3}c^{5}d^{10}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5E-2 | $5$ | $1331$ | $C_5\times C_{40}$ | 5E1 | 1A | 5E-2 | $a^{16}b^{7}c^{8}d^{5}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5F1 | $5$ | $1331$ | $C_5\times C_{40}$ | 5F2 | 1A | 5F1 | $a^{32}b^{9}cd^{22}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5F-1 | $5$ | $1331$ | $C_5\times C_{40}$ | 5F-2 | 1A | 5F-1 | $a^{8}b^{8}c^{7}d^{33}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5F2 | $5$ | $1331$ | $C_5\times C_{40}$ | 5F-1 | 1A | 5F2 | $a^{24}b^{3}c^{4}d^{44}$ |
| $C_5\times C_{11}^3:C_{40}$ | 5F-2 | $5$ | $1331$ | $C_5\times C_{40}$ | 5F1 | 1A | 5F-2 | $a^{16}b^{7}c^{2}d^{11}$ |
| $C_5\times C_{11}^3:C_{40}$ | 8A1 | $8$ | $1331$ | $C_5\times C_{40}$ | 4A1 | 8A-3 | 8A3 | $a^{5}b^{3}c^{4}d^{15}$ |
| $C_5\times C_{11}^3:C_{40}$ | 8A-1 | $8$ | $1331$ | $C_5\times C_{40}$ | 4A-1 | 8A3 | 8A-3 | $a^{35}c^{5}d^{30}$ |
| $C_5\times C_{11}^3:C_{40}$ | 8A3 | $8$ | $1331$ | $C_5\times C_{40}$ | 4A-1 | 8A-1 | 8A1 | $a^{15}c^{5}d^{30}$ |
| $C_5\times C_{11}^3:C_{40}$ | 8A-3 | $8$ | $1331$ | $C_5\times C_{40}$ | 4A1 | 8A1 | 8A-1 | $a^{25}b^{3}c^{4}d^{15}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10A1 | $10$ | $1$ | $C_5\times C_{11}^3:C_{40}$ | 5A1 | 2A | 10A1 | $a^{20}d^{11}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10A-1 | $10$ | $1$ | $C_5\times C_{11}^3:C_{40}$ | 5A-1 | 2A | 10A-1 | $a^{20}d^{44}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10A3 | $10$ | $1$ | $C_5\times C_{11}^3:C_{40}$ | 5A-2 | 2A | 10A3 | $a^{20}d^{33}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10A-3 | $10$ | $1$ | $C_5\times C_{11}^3:C_{40}$ | 5A2 | 2A | 10A-3 | $a^{20}d^{22}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10B1 | $10$ | $1331$ | $C_5\times C_{40}$ | 5B1 | 2A | 10B1 | $a^{12}b^{8}c^{2}d^{54}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10B-1 | $10$ | $1331$ | $C_5\times C_{40}$ | 5B-1 | 2A | 10B-1 | $a^{28}bc^{3}d^{26}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10B3 | $10$ | $1331$ | $C_5\times C_{40}$ | 5B-2 | 2A | 10B3 | $a^{36}b^{5}c^{4}d^{42}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10B-3 | $10$ | $1331$ | $C_5\times C_{40}$ | 5B2 | 2A | 10B-3 | $a^{4}b^{10}c^{8}d^{18}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10C1 | $10$ | $1331$ | $C_5\times C_{40}$ | 5C1 | 2A | 10C1 | $a^{4}b^{8}c^{3}d^{32}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10C-1 | $10$ | $1331$ | $C_5\times C_{40}$ | 5C-1 | 2A | 10C-1 | $a^{36}b^{4}c^{7}d^{38}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10C3 | $10$ | $1331$ | $C_5\times C_{40}$ | 5C-2 | 2A | 10C3 | $a^{12}b^{2}c^{9}d^{41}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10C-3 | $10$ | $1331$ | $C_5\times C_{40}$ | 5C2 | 2A | 10C-3 | $a^{28}b^{3}c^{8}d^{34}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10D1 | $10$ | $1331$ | $C_5\times C_{40}$ | 5D1 | 2A | 10D1 | $a^{12}b^{8}c^{2}d^{23}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10D-1 | $10$ | $1331$ | $C_5\times C_{40}$ | 5D-1 | 2A | 10D-1 | $a^{28}bc^{3}d^{7}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10D3 | $10$ | $1331$ | $C_5\times C_{40}$ | 5D-2 | 2A | 10D3 | $a^{36}b^{5}c^{4}d^{24}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10D-3 | $10$ | $1331$ | $C_5\times C_{40}$ | 5D2 | 2A | 10D-3 | $a^{4}b^{10}c^{8}d^{26}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10E1 | $10$ | $1331$ | $C_5\times C_{40}$ | 5E1 | 2A | 10E1 | $a^{36}b^{7}c^{8}d^{5}$ |
| $C_5\times C_{11}^3:C_{40}$ | 10E-1 | $10$ | $1331$ | $C_5\times C_{40}$ | 5E-1 | 2A | 10E-1 | $a^{4}b^{3}c^{5}d^{10}$ |