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Elements of the group are displayed as words in the presentation $\langle a, b \mid a^{2}=b^{62250}=1, b^{a}=b^{499} \rangle$ .
| Group | Label | Order | Size | Centralizer | Powers | Representative | |||
|---|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 5P | 83P | ||||||
| $C_{249}\times D_{250}$ | 1A | $1$ | $1$ | not computed | 1A | 1A | 1A | 1A | $1$ |
| $C_{249}\times D_{250}$ | 2A | $2$ | $1$ | not computed | 1A | 2A | 2A | 2A | $b^{31125}$ |
| $C_{249}\times D_{250}$ | 2B | $2$ | $125$ | not computed | 1A | 2B | 2B | 2B | $ab^{61752}$ |
| $C_{249}\times D_{250}$ | 2C | $2$ | $125$ | not computed | 1A | 2C | 2C | 2C | $ab^{62001}$ |
| $C_{249}\times D_{250}$ | 3A1 | $3$ | $1$ | not computed | 3A-1 | 1A | 3A-1 | 3A-1 | $b^{20750}$ |
| $C_{249}\times D_{250}$ | 3A-1 | $3$ | $1$ | not computed | 3A1 | 1A | 3A1 | 3A1 | $b^{41500}$ |
| $C_{249}\times D_{250}$ | 5A1 | $5$ | $2$ | not computed | 5A2 | 5A2 | 1A | 5A1 | $b^{12450}$ |
| $C_{249}\times D_{250}$ | 5A2 | $5$ | $2$ | not computed | 5A1 | 5A1 | 1A | 5A2 | $b^{24900}$ |
| $C_{249}\times D_{250}$ | 6A1 | $6$ | $1$ | not computed | 3A1 | 2A | 6A-1 | 6A-1 | $b^{10375}$ |
| $C_{249}\times D_{250}$ | 6A-1 | $6$ | $1$ | not computed | 3A-1 | 2A | 6A1 | 6A1 | $b^{51875}$ |
| $C_{249}\times D_{250}$ | 6B1 | $6$ | $125$ | not computed | 3A1 | 2B | 6B-1 | 6B-1 | $ab^{41002}$ |
| $C_{249}\times D_{250}$ | 6B-1 | $6$ | $125$ | not computed | 3A-1 | 2B | 6B1 | 6B1 | $ab^{20252}$ |
| $C_{249}\times D_{250}$ | 6C1 | $6$ | $125$ | not computed | 3A-1 | 2C | 6C-1 | 6C-1 | $ab^{20501}$ |
| $C_{249}\times D_{250}$ | 6C-1 | $6$ | $125$ | not computed | 3A1 | 2C | 6C1 | 6C1 | $ab^{41251}$ |
| $C_{249}\times D_{250}$ | 10A1 | $10$ | $2$ | not computed | 5A1 | 10A3 | 2A | 10A1 | $b^{6225}$ |
| $C_{249}\times D_{250}$ | 10A3 | $10$ | $2$ | not computed | 5A2 | 10A1 | 2A | 10A3 | $b^{18675}$ |
| $C_{249}\times D_{250}$ | 15A1 | $15$ | $2$ | not computed | 15A2 | 5A1 | 3A1 | 15A-1 | $b^{4150}$ |
| $C_{249}\times D_{250}$ | 15A-1 | $15$ | $2$ | not computed | 15A-2 | 5A1 | 3A-1 | 15A1 | $b^{58100}$ |
| $C_{249}\times D_{250}$ | 15A2 | $15$ | $2$ | not computed | 15A1 | 5A2 | 3A-1 | 15A-2 | $b^{8300}$ |
| $C_{249}\times D_{250}$ | 15A-2 | $15$ | $2$ | not computed | 15A-1 | 5A2 | 3A1 | 15A2 | $b^{53950}$ |
| $C_{249}\times D_{250}$ | 25A1 | $25$ | $2$ | not computed | 25A2 | 25A3 | 5A1 | 25A9 | $b^{2490}$ |
| $C_{249}\times D_{250}$ | 25A2 | $25$ | $2$ | not computed | 25A4 | 25A6 | 5A2 | 25A7 | $b^{4980}$ |
| $C_{249}\times D_{250}$ | 25A3 | $25$ | $2$ | not computed | 25A6 | 25A9 | 5A2 | 25A2 | $b^{7470}$ |
| $C_{249}\times D_{250}$ | 25A4 | $25$ | $2$ | not computed | 25A8 | 25A12 | 5A1 | 25A11 | $b^{9960}$ |
| $C_{249}\times D_{250}$ | 25A6 | $25$ | $2$ | not computed | 25A12 | 25A7 | 5A1 | 25A4 | $b^{14940}$ |
| $C_{249}\times D_{250}$ | 25A7 | $25$ | $2$ | not computed | 25A11 | 25A4 | 5A2 | 25A12 | $b^{17430}$ |
| $C_{249}\times D_{250}$ | 25A8 | $25$ | $2$ | not computed | 25A9 | 25A1 | 5A2 | 25A3 | $b^{19920}$ |
| $C_{249}\times D_{250}$ | 25A9 | $25$ | $2$ | not computed | 25A7 | 25A2 | 5A1 | 25A6 | $b^{22410}$ |
| $C_{249}\times D_{250}$ | 25A11 | $25$ | $2$ | not computed | 25A3 | 25A8 | 5A1 | 25A1 | $b^{27390}$ |
| $C_{249}\times D_{250}$ | 25A12 | $25$ | $2$ | not computed | 25A1 | 25A11 | 5A2 | 25A8 | $b^{29880}$ |
| $C_{249}\times D_{250}$ | 30A1 | $30$ | $2$ | not computed | 15A1 | 10A1 | 6A1 | 30A-1 | $b^{2075}$ |
| $C_{249}\times D_{250}$ | 30A-1 | $30$ | $2$ | not computed | 15A-1 | 10A1 | 6A-1 | 30A1 | $b^{60175}$ |
| $C_{249}\times D_{250}$ | 30A7 | $30$ | $2$ | not computed | 15A-2 | 10A3 | 6A1 | 30A-7 | $b^{14525}$ |
| $C_{249}\times D_{250}$ | 30A-7 | $30$ | $2$ | not computed | 15A2 | 10A3 | 6A-1 | 30A7 | $b^{47725}$ |
| $C_{249}\times D_{250}$ | 50A1 | $50$ | $2$ | not computed | 25A1 | 50A3 | 10A1 | 50A9 | $b^{1245}$ |
| $C_{249}\times D_{250}$ | 50A3 | $50$ | $2$ | not computed | 25A3 | 50A9 | 10A3 | 50A23 | $b^{3735}$ |
| $C_{249}\times D_{250}$ | 50A7 | $50$ | $2$ | not computed | 25A7 | 50A21 | 10A3 | 50A13 | $b^{8715}$ |
| $C_{249}\times D_{250}$ | 50A9 | $50$ | $2$ | not computed | 25A9 | 50A23 | 10A1 | 50A19 | $b^{11205}$ |
| $C_{249}\times D_{250}$ | 50A11 | $50$ | $2$ | not computed | 25A11 | 50A17 | 10A1 | 50A1 | $b^{13695}$ |
| $C_{249}\times D_{250}$ | 50A13 | $50$ | $2$ | not computed | 25A12 | 50A11 | 10A3 | 50A17 | $b^{16185}$ |
| $C_{249}\times D_{250}$ | 50A17 | $50$ | $2$ | not computed | 25A8 | 50A1 | 10A3 | 50A3 | $b^{21165}$ |
| $C_{249}\times D_{250}$ | 50A19 | $50$ | $2$ | not computed | 25A6 | 50A7 | 10A1 | 50A21 | $b^{23655}$ |
| $C_{249}\times D_{250}$ | 50A21 | $50$ | $2$ | not computed | 25A4 | 50A13 | 10A1 | 50A11 | $b^{26145}$ |
| $C_{249}\times D_{250}$ | 50A23 | $50$ | $2$ | not computed | 25A2 | 50A19 | 10A3 | 50A7 | $b^{28635}$ |
| $C_{249}\times D_{250}$ | 75A1 | $75$ | $2$ | not computed | 75A2 | 25A1 | 15A1 | 75A-16 | $b^{830}$ |
| $C_{249}\times D_{250}$ | 75A-1 | $75$ | $2$ | not computed | 75A-2 | 25A1 | 15A-1 | 75A16 | $b^{61420}$ |
| $C_{249}\times D_{250}$ | 75A2 | $75$ | $2$ | not computed | 75A4 | 25A2 | 15A2 | 75A7 | $b^{1660}$ |
| $C_{249}\times D_{250}$ | 75A-2 | $75$ | $2$ | not computed | 75A-4 | 25A2 | 15A-2 | 75A-7 | $b^{60590}$ |
| $C_{249}\times D_{250}$ | 75A4 | $75$ | $2$ | not computed | 75A8 | 25A4 | 15A1 | 75A11 | $b^{3320}$ |
| $C_{249}\times D_{250}$ | 75A-4 | $75$ | $2$ | not computed | 75A-8 | 25A4 | 15A-1 | 75A-11 | $b^{58930}$ |