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Elements of the group are displayed as matrices in $\GL_{2}(\Z/{85}\Z)$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | ||
|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 5P | ||||||
| $C_2\times C_{20}.S_4$ | 1A | $1$ | $1$ | $C_2\times C_{20}.S_4$ | 1A | 1A | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 2A | $2$ | $1$ | $C_2\times C_{20}.S_4$ | 1A | 2A | 2A | $ \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 2B | $2$ | $1$ | $C_2\times C_{20}.S_4$ | 1A | 2B | 2B | $ \left(\begin{array}{rr} 69 & 0 \\ 0 & 69 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 2C | $2$ | $1$ | $C_2\times C_{20}.S_4$ | 1A | 2C | 2C | $ \left(\begin{array}{rr} 84 & 0 \\ 0 & 84 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 2D | $2$ | $6$ | $C_{20}.C_2^3$ | 1A | 2D | 2D | $ \left(\begin{array}{rr} 51 & 15 \\ 25 & 51 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 2E | $2$ | $6$ | $C_{20}.C_2^3$ | 1A | 2E | 2E | $ \left(\begin{array}{rr} 34 & 15 \\ 25 & 34 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 2F | $2$ | $60$ | $C_2^2\times C_4$ | 1A | 2F | 2F | $ \left(\begin{array}{rr} 34 & 67 \\ 50 & 51 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 2G | $2$ | $60$ | $C_2^2\times C_4$ | 1A | 2G | 2G | $ \left(\begin{array}{rr} 71 & 79 \\ 75 & 14 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 3A | $3$ | $8$ | $C_2\times C_{60}$ | 3A | 1A | 3A | $ \left(\begin{array}{rr} 6 & 5 \\ 5 & 61 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 4A1 | $4$ | $1$ | $C_2\times C_{20}.S_4$ | 2A | 4A-1 | 4A1 | $ \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 4A-1 | $4$ | $1$ | $C_2\times C_{20}.S_4$ | 2A | 4A1 | 4A-1 | $ \left(\begin{array}{rr} 81 & 0 \\ 0 & 81 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 4B1 | $4$ | $1$ | $C_2\times C_{20}.S_4$ | 2A | 4B-1 | 4B1 | $ \left(\begin{array}{rr} 64 & 0 \\ 0 & 64 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 4B-1 | $4$ | $1$ | $C_2\times C_{20}.S_4$ | 2A | 4B1 | 4B-1 | $ \left(\begin{array}{rr} 4 & 0 \\ 0 & 4 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 4C | $4$ | $6$ | $C_{20}.C_2^3$ | 2A | 4C | 4C | $ \left(\begin{array}{rr} 21 & 0 \\ 0 & 81 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 4D | $4$ | $6$ | $C_{20}.C_2^3$ | 2A | 4D | 4D | $ \left(\begin{array}{rr} 34 & 15 \\ 60 & 34 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 4E | $4$ | $60$ | $C_2^2\times C_4$ | 2A | 4E | 4E | $ \left(\begin{array}{rr} 51 & 50 \\ 35 & 34 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 4F | $4$ | $60$ | $C_2^2\times C_4$ | 2A | 4F | 4F | $ \left(\begin{array}{rr} 34 & 47 \\ 30 & 51 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 5A1 | $5$ | $2$ | $C_2\times \SL(2,3):C_{10}$ | 5A2 | 5A2 | 1A | $ \left(\begin{array}{rr} 1 & 51 \\ 0 & 1 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 5A2 | $5$ | $2$ | $C_2\times \SL(2,3):C_{10}$ | 5A1 | 5A1 | 1A | $ \left(\begin{array}{rr} 1 & 17 \\ 0 & 1 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 6A | $6$ | $8$ | $C_2\times C_{60}$ | 3A | 2B | 6A | $ \left(\begin{array}{rr} 44 & 80 \\ 80 & 74 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 6B | $6$ | $8$ | $C_2\times C_{60}$ | 3A | 2C | 6B | $ \left(\begin{array}{rr} 24 & 5 \\ 5 & 79 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 6C | $6$ | $8$ | $C_2\times C_{60}$ | 3A | 2A | 6C | $ \left(\begin{array}{rr} 41 & 80 \\ 80 & 11 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 8A1 | $8$ | $30$ | $C_2^2\times C_8$ | 4C | 8A1 | 8A-1 | $ \left(\begin{array}{rr} 66 & 17 \\ 0 & 9 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 8A-1 | $8$ | $30$ | $C_2^2\times C_8$ | 4C | 8A-1 | 8A1 | $ \left(\begin{array}{rr} 76 & 0 \\ 0 & 19 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 8B1 | $8$ | $30$ | $C_2^2\times C_8$ | 4C | 8B3 | 8B3 | $ \left(\begin{array}{rr} 36 & 0 \\ 0 & 9 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 8B3 | $8$ | $30$ | $C_2^2\times C_8$ | 4C | 8B1 | 8B1 | $ \left(\begin{array}{rr} 66 & 34 \\ 0 & 59 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 8C1 | $8$ | $30$ | $C_2^2\times C_8$ | 4C | 8C3 | 8C3 | $ \left(\begin{array}{rr} 54 & 40 \\ 75 & 71 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 8C3 | $8$ | $30$ | $C_2^2\times C_8$ | 4C | 8C1 | 8C1 | $ \left(\begin{array}{rr} 14 & 11 \\ 10 & 31 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 8D1 | $8$ | $30$ | $C_2^2\times C_8$ | 4C | 8D1 | 8D-1 | $ \left(\begin{array}{rr} 29 & 75 \\ 45 & 46 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 8D-1 | $8$ | $30$ | $C_2^2\times C_8$ | 4C | 8D-1 | 8D1 | $ \left(\begin{array}{rr} 39 & 61 \\ 40 & 56 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 10A1 | $10$ | $2$ | $C_2\times \SL(2,3):C_{10}$ | 5A2 | 10A3 | 2A | $ \left(\begin{array}{rr} 16 & 34 \\ 0 & 16 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 10A3 | $10$ | $2$ | $C_2\times \SL(2,3):C_{10}$ | 5A1 | 10A1 | 2A | $ \left(\begin{array}{rr} 16 & 17 \\ 0 & 16 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 10B1 | $10$ | $2$ | $C_2\times \SL(2,3):C_{10}$ | 5A2 | 10B3 | 2B | $ \left(\begin{array}{rr} 69 & 51 \\ 0 & 69 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 10B3 | $10$ | $2$ | $C_2\times \SL(2,3):C_{10}$ | 5A1 | 10B1 | 2B | $ \left(\begin{array}{rr} 69 & 68 \\ 0 & 69 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 10C1 | $10$ | $2$ | $C_2\times \SL(2,3):C_{10}$ | 5A1 | 10C3 | 2C | $ \left(\begin{array}{rr} 84 & 68 \\ 0 & 84 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 10C3 | $10$ | $2$ | $C_2\times \SL(2,3):C_{10}$ | 5A2 | 10C1 | 2C | $ \left(\begin{array}{rr} 84 & 34 \\ 0 & 84 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 10D1 | $10$ | $12$ | $C_2^2\times C_{20}$ | 5A1 | 10D3 | 2D | $ \left(\begin{array}{rr} 51 & 83 \\ 25 & 51 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 10D3 | $10$ | $12$ | $C_2^2\times C_{20}$ | 5A2 | 10D1 | 2D | $ \left(\begin{array}{rr} 51 & 19 \\ 60 & 51 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 10E1 | $10$ | $12$ | $C_2^2\times C_{20}$ | 5A1 | 10E3 | 2E | $ \left(\begin{array}{rr} 34 & 32 \\ 25 & 34 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 10E3 | $10$ | $12$ | $C_2^2\times C_{20}$ | 5A2 | 10E1 | 2E | $ \left(\begin{array}{rr} 34 & 36 \\ 60 & 34 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 12A1 | $12$ | $8$ | $C_2\times C_{60}$ | 6C | 4A1 | 12A1 | $ \left(\begin{array}{rr} 61 & 20 \\ 20 & 11 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 12A-1 | $12$ | $8$ | $C_2\times C_{60}$ | 6C | 4A-1 | 12A-1 | $ \left(\begin{array}{rr} 6 & 20 \\ 20 & 41 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 12B1 | $12$ | $8$ | $C_2\times C_{60}$ | 6C | 4B1 | 12B1 | $ \left(\begin{array}{rr} 24 & 80 \\ 5 & 74 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 12B-1 | $12$ | $8$ | $C_2\times C_{60}$ | 6C | 4B-1 | 12B-1 | $ \left(\begin{array}{rr} 79 & 80 \\ 5 & 44 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 15A1 | $15$ | $8$ | $C_2\times C_{60}$ | 15A2 | 5A2 | 3A | $ \left(\begin{array}{rr} 61 & 46 \\ 80 & 6 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 15A2 | $15$ | $8$ | $C_2\times C_{60}$ | 15A4 | 5A1 | 3A | $ \left(\begin{array}{rr} 6 & 22 \\ 5 & 61 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 15A4 | $15$ | $8$ | $C_2\times C_{60}$ | 15A7 | 5A2 | 3A | $ \left(\begin{array}{rr} 61 & 29 \\ 80 & 6 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 15A7 | $15$ | $8$ | $C_2\times C_{60}$ | 15A1 | 5A1 | 3A | $ \left(\begin{array}{rr} 61 & 12 \\ 80 & 6 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 20A1 | $20$ | $2$ | $C_2\times \SL(2,3):C_{10}$ | 10A3 | 20A3 | 4A1 | $ \left(\begin{array}{rr} 21 & 34 \\ 0 & 21 \end{array}\right) $ |
| $C_2\times C_{20}.S_4$ | 20A-1 | $20$ | $2$ | $C_2\times \SL(2,3):C_{10}$ | 10A3 | 20A-3 | 4A-1 | $ \left(\begin{array}{rr} 81 & 51 \\ 0 & 81 \end{array}\right) $ |