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Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\SL(2,113)$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | ||||
|---|---|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 7P | 19P | 113P | ||||||
| $\PSL(2,113)$ | 1A | $1$ | $1$ | $\PSL(2,113)$ | 1A | 1A | 1A | 1A | 1A | $ \left[ \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 2A | $2$ | $6441$ | $D_{56}$ | 1A | 2A | 2A | 2A | 2A | $ \left[ \left(\begin{array}{rr} 81 & 98 \\ 106 & 32 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 3A | $3$ | $12656$ | $C_{57}$ | 3A | 1A | 3A | 3A | 3A | $ \left[ \left(\begin{array}{rr} 15 & 106 \\ 99 & 97 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 4A | $4$ | $12882$ | $C_{56}$ | 2A | 4A | 4A | 4A | 4A | $ \left[ \left(\begin{array}{rr} 107 & 100 \\ 9 & 57 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 7A1 | $7$ | $12882$ | $C_{56}$ | 7A2 | 7A3 | 1A | 7A2 | 7A1 | $ \left[ \left(\begin{array}{rr} 105 & 16 \\ 15 & 97 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 7A2 | $7$ | $12882$ | $C_{56}$ | 7A3 | 7A1 | 1A | 7A3 | 7A2 | $ \left[ \left(\begin{array}{rr} 35 & 45 \\ 21 & 69 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 7A3 | $7$ | $12882$ | $C_{56}$ | 7A1 | 7A2 | 1A | 7A1 | 7A3 | $ \left[ \left(\begin{array}{rr} 57 & 47 \\ 37 & 90 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 8A1 | $8$ | $12882$ | $C_{56}$ | 4A | 8A3 | 8A1 | 8A3 | 8A1 | $ \left[ \left(\begin{array}{rr} 98 & 69 \\ 100 & 7 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 8A3 | $8$ | $12882$ | $C_{56}$ | 4A | 8A1 | 8A3 | 8A1 | 8A3 | $ \left[ \left(\begin{array}{rr} 80 & 53 \\ 85 & 110 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 14A1 | $14$ | $12882$ | $C_{56}$ | 7A1 | 14A3 | 2A | 14A5 | 14A1 | $ \left[ \left(\begin{array}{rr} 65 & 29 \\ 6 & 107 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 14A3 | $14$ | $12882$ | $C_{56}$ | 7A3 | 14A5 | 2A | 14A1 | 14A3 | $ \left[ \left(\begin{array}{rr} 85 & 102 \\ 25 & 34 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 14A5 | $14$ | $12882$ | $C_{56}$ | 7A2 | 14A1 | 2A | 14A3 | 14A5 | $ \left[ \left(\begin{array}{rr} 54 & 46 \\ 29 & 31 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 19A1 | $19$ | $12656$ | $C_{57}$ | 19A2 | 19A3 | 19A7 | 1A | 19A1 | $ \left[ \left(\begin{array}{rr} 25 & 63 \\ 13 & 78 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 19A2 | $19$ | $12656$ | $C_{57}$ | 19A4 | 19A6 | 19A5 | 1A | 19A2 | $ \left[ \left(\begin{array}{rr} 25 & 65 \\ 17 & 103 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 19A3 | $19$ | $12656$ | $C_{57}$ | 19A6 | 19A9 | 19A2 | 1A | 19A3 | $ \left[ \left(\begin{array}{rr} 112 & 22 \\ 44 & 48 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 19A4 | $19$ | $12656$ | $C_{57}$ | 19A8 | 19A7 | 19A9 | 1A | 19A4 | $ \left[ \left(\begin{array}{rr} 35 & 71 \\ 29 & 75 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 19A5 | $19$ | $12656$ | $C_{57}$ | 19A9 | 19A4 | 19A3 | 1A | 19A5 | $ \left[ \left(\begin{array}{rr} 103 & 59 \\ 5 & 106 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 19A6 | $19$ | $12656$ | $C_{57}$ | 19A7 | 19A1 | 19A4 | 1A | 19A6 | $ \left[ \left(\begin{array}{rr} 48 & 96 \\ 79 & 5 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 19A7 | $19$ | $12656$ | $C_{57}$ | 19A5 | 19A2 | 19A8 | 1A | 19A7 | $ \left[ \left(\begin{array}{rr} 38 & 110 \\ 107 & 57 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 19A8 | $19$ | $12656$ | $C_{57}$ | 19A3 | 19A5 | 19A1 | 1A | 19A8 | $ \left[ \left(\begin{array}{rr} 7 & 13 \\ 26 & 0 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 19A9 | $19$ | $12656$ | $C_{57}$ | 19A1 | 19A8 | 19A6 | 1A | 19A9 | $ \left[ \left(\begin{array}{rr} 5 & 99 \\ 85 & 56 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 28A1 | $28$ | $12882$ | $C_{56}$ | 14A1 | 28A3 | 4A | 28A9 | 28A1 | $ \left[ \left(\begin{array}{rr} 105 & 41 \\ 102 & 28 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 28A3 | $28$ | $12882$ | $C_{56}$ | 14A3 | 28A9 | 4A | 28A1 | 28A3 | $ \left[ \left(\begin{array}{rr} 65 & 87 \\ 18 & 78 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 28A5 | $28$ | $12882$ | $C_{56}$ | 14A5 | 28A13 | 4A | 28A11 | 28A5 | $ \left[ \left(\begin{array}{rr} 1 & 7 \\ 56 & 54 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 28A9 | $28$ | $12882$ | $C_{56}$ | 14A5 | 28A1 | 4A | 28A3 | 28A9 | $ \left[ \left(\begin{array}{rr} 97 & 17 \\ 23 & 32 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 28A11 | $28$ | $12882$ | $C_{56}$ | 14A3 | 28A5 | 4A | 28A13 | 28A11 | $ \left[ \left(\begin{array}{rr} 79 & 112 \\ 105 & 23 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 28A13 | $28$ | $12882$ | $C_{56}$ | 14A1 | 28A11 | 4A | 28A5 | 28A13 | $ \left[ \left(\begin{array}{rr} 69 & 76 \\ 43 & 31 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 56A1 | $56$ | $12882$ | $C_{56}$ | 28A1 | 56A3 | 8A1 | 56A19 | 56A1 | $ \left[ \left(\begin{array}{rr} 74 & 27 \\ 103 & 4 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 56A3 | $56$ | $12882$ | $C_{56}$ | 28A3 | 56A9 | 8A3 | 56A1 | 56A3 | $ \left[ \left(\begin{array}{rr} 15 & 61 \\ 36 & 41 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 56A5 | $56$ | $12882$ | $C_{56}$ | 28A5 | 56A15 | 8A3 | 56A17 | 56A5 | $ \left[ \left(\begin{array}{rr} 0 & 63 \\ 52 & 25 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 56A9 | $56$ | $12882$ | $C_{56}$ | 28A9 | 56A27 | 8A1 | 56A3 | 56A9 | $ \left[ \left(\begin{array}{rr} 74 & 39 \\ 86 & 111 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 56A11 | $56$ | $12882$ | $C_{56}$ | 28A11 | 56A23 | 8A3 | 56A15 | 56A11 | $ \left[ \left(\begin{array}{rr} 4 & 55 \\ 101 & 33 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 56A13 | $56$ | $12882$ | $C_{56}$ | 28A13 | 56A17 | 8A3 | 56A23 | 56A13 | $ \left[ \left(\begin{array}{rr} 72 & 104 \\ 41 & 20 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 56A15 | $56$ | $12882$ | $C_{56}$ | 28A13 | 56A11 | 8A1 | 56A5 | 56A15 | $ \left[ \left(\begin{array}{rr} 88 & 101 \\ 17 & 94 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 56A17 | $56$ | $12882$ | $C_{56}$ | 28A11 | 56A5 | 8A1 | 56A13 | 56A17 | $ \left[ \left(\begin{array}{rr} 106 & 108 \\ 73 & 52 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 56A19 | $56$ | $12882$ | $C_{56}$ | 28A9 | 56A1 | 8A3 | 56A25 | 56A19 | $ \left[ \left(\begin{array}{rr} 2 & 88 \\ 26 & 71 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 56A23 | $56$ | $12882$ | $C_{56}$ | 28A5 | 56A13 | 8A1 | 56A11 | 56A23 | $ \left[ \left(\begin{array}{rr} 93 & 95 \\ 82 & 102 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 56A25 | $56$ | $12882$ | $C_{56}$ | 28A3 | 56A19 | 8A1 | 56A27 | 56A25 | $ \left[ \left(\begin{array}{rr} 94 & 81 \\ 83 & 110 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 56A27 | $56$ | $12882$ | $C_{56}$ | 28A1 | 56A25 | 8A3 | 56A9 | 56A27 | $ \left[ \left(\begin{array}{rr} 52 & 20 \\ 47 & 42 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 57A1 | $57$ | $12656$ | $C_{57}$ | 57A2 | 19A1 | 57A7 | 3A | 57A1 | $ \left[ \left(\begin{array}{rr} 63 & 32 \\ 64 & 11 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 57A2 | $57$ | $12656$ | $C_{57}$ | 57A4 | 19A2 | 57A14 | 3A | 57A2 | $ \left[ \left(\begin{array}{rr} 85 & 5 \\ 10 & 91 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 57A4 | $57$ | $12656$ | $C_{57}$ | 57A8 | 19A4 | 57A28 | 3A | 57A4 | $ \left[ \left(\begin{array}{rr} 70 & 24 \\ 48 & 31 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 57A5 | $57$ | $12656$ | $C_{57}$ | 57A10 | 19A5 | 57A22 | 3A | 57A5 | $ \left[ \left(\begin{array}{rr} 70 & 18 \\ 36 & 69 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 57A7 | $57$ | $12656$ | $C_{57}$ | 57A14 | 19A7 | 57A8 | 3A | 57A7 | $ \left[ \left(\begin{array}{rr} 28 & 67 \\ 21 & 18 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 57A8 | $57$ | $12656$ | $C_{57}$ | 57A16 | 19A8 | 57A1 | 3A | 57A8 | $ \left[ \left(\begin{array}{rr} 63 & 51 \\ 102 & 79 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 57A10 | $57$ | $12656$ | $C_{57}$ | 57A20 | 19A9 | 57A13 | 3A | 57A10 | $ \left[ \left(\begin{array}{rr} 11 & 16 \\ 32 & 98 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 57A11 | $57$ | $12656$ | $C_{57}$ | 57A22 | 19A8 | 57A20 | 3A | 57A11 | $ \left[ \left(\begin{array}{rr} 22 & 76 \\ 39 & 68 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 57A13 | $57$ | $12656$ | $C_{57}$ | 57A26 | 19A6 | 57A23 | 3A | 57A13 | $ \left[ \left(\begin{array}{rr} 31 & 20 \\ 40 & 55 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 57A14 | $57$ | $12656$ | $C_{57}$ | 57A28 | 19A5 | 57A16 | 3A | 57A14 | $ \left[ \left(\begin{array}{rr} 44 & 31 \\ 62 & 36 \end{array}\right) \right] $ |
| $\PSL(2,113)$ | 57A16 | $57$ | $12656$ | $C_{57}$ | 57A25 | 19A3 | 57A2 | 3A | 57A16 | $ \left[ \left(\begin{array}{rr} 95 & 103 \\ 93 & 83 \end{array}\right) \right] $ |