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Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\SL(2,127)$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |||
|---|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 7P | 127P | ||||||
| $\PSL(2,127)$ | 1A | $1$ | $1$ | $\PSL(2,127)$ | 1A | 1A | 1A | 1A | $ \left[ \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 2A | $2$ | $8001$ | $D_{64}$ | 1A | 2A | 2A | 2A | $ \left[ \left(\begin{array}{rr} 33 & 118 \\ 107 & 94 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 3A | $3$ | $16256$ | $C_{63}$ | 3A | 1A | 3A | 3A | $ \left[ \left(\begin{array}{rr} 105 & 123 \\ 84 & 21 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 4A | $4$ | $16002$ | $C_{64}$ | 2A | 4A | 4A | 4A | $ \left[ \left(\begin{array}{rr} 2 & 55 \\ 94 & 109 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 7A1 | $7$ | $16256$ | $C_{63}$ | 7A2 | 7A3 | 1A | 7A1 | $ \left[ \left(\begin{array}{rr} 89 & 35 \\ 27 & 62 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 7A2 | $7$ | $16256$ | $C_{63}$ | 7A3 | 7A1 | 1A | 7A2 | $ \left[ \left(\begin{array}{rr} 24 & 49 \\ 114 & 37 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 7A3 | $7$ | $16256$ | $C_{63}$ | 7A1 | 7A2 | 1A | 7A3 | $ \left[ \left(\begin{array}{rr} 97 & 59 \\ 31 & 66 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 8A1 | $8$ | $16002$ | $C_{64}$ | 4A | 8A3 | 8A1 | 8A1 | $ \left[ \left(\begin{array}{rr} 9 & 38 \\ 28 & 76 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 8A3 | $8$ | $16002$ | $C_{64}$ | 4A | 8A1 | 8A3 | 8A3 | $ \left[ \left(\begin{array}{rr} 34 & 65 \\ 88 & 45 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 9A1 | $9$ | $16256$ | $C_{63}$ | 9A2 | 3A | 9A2 | 9A1 | $ \left[ \left(\begin{array}{rr} 102 & 3 \\ 64 & 38 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 9A2 | $9$ | $16256$ | $C_{63}$ | 9A4 | 3A | 9A4 | 9A2 | $ \left[ \left(\begin{array}{rr} 55 & 39 \\ 70 & 112 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 9A4 | $9$ | $16256$ | $C_{63}$ | 9A1 | 3A | 9A1 | 9A4 | $ \left[ \left(\begin{array}{rr} 40 & 36 \\ 6 & 34 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 16A1 | $16$ | $16002$ | $C_{64}$ | 8A1 | 16A3 | 16A7 | 16A1 | $ \left[ \left(\begin{array}{rr} 34 & 98 \\ 119 & 33 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 16A3 | $16$ | $16002$ | $C_{64}$ | 8A3 | 16A7 | 16A5 | 16A3 | $ \left[ \left(\begin{array}{rr} 2 & 104 \\ 90 & 45 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 16A5 | $16$ | $16002$ | $C_{64}$ | 8A3 | 16A1 | 16A3 | 16A5 | $ \left[ \left(\begin{array}{rr} 118 & 21 \\ 89 & 18 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 16A7 | $16$ | $16002$ | $C_{64}$ | 8A1 | 16A5 | 16A1 | 16A7 | $ \left[ \left(\begin{array}{rr} 1 & 16 \\ 92 & 76 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 21A1 | $21$ | $16256$ | $C_{63}$ | 21A2 | 7A1 | 3A | 21A1 | $ \left[ \left(\begin{array}{rr} 23 & 93 \\ 79 & 71 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 21A2 | $21$ | $16256$ | $C_{63}$ | 21A4 | 7A2 | 3A | 21A2 | $ \left[ \left(\begin{array}{rr} 2 & 106 \\ 60 & 69 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 21A4 | $21$ | $16256$ | $C_{63}$ | 21A8 | 7A3 | 3A | 21A4 | $ \left[ \left(\begin{array}{rr} 113 & 94 \\ 58 & 55 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 21A5 | $21$ | $16256$ | $C_{63}$ | 21A10 | 7A2 | 3A | 21A5 | $ \left[ \left(\begin{array}{rr} 119 & 38 \\ 91 & 28 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 21A8 | $21$ | $16256$ | $C_{63}$ | 21A5 | 7A1 | 3A | 21A8 | $ \left[ \left(\begin{array}{rr} 67 & 83 \\ 35 & 32 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 21A10 | $21$ | $16256$ | $C_{63}$ | 21A1 | 7A3 | 3A | 21A10 | $ \left[ \left(\begin{array}{rr} 93 & 125 \\ 42 & 51 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 32A1 | $32$ | $16002$ | $C_{64}$ | 16A1 | 32A3 | 32A7 | 32A1 | $ \left[ \left(\begin{array}{rr} 121 & 63 \\ 13 & 75 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 32A3 | $32$ | $16002$ | $C_{64}$ | 16A3 | 32A9 | 32A11 | 32A3 | $ \left[ \left(\begin{array}{rr} 54 & 94 \\ 96 & 66 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 32A5 | $32$ | $16002$ | $C_{64}$ | 16A5 | 32A15 | 32A3 | 32A5 | $ \left[ \left(\begin{array}{rr} 59 & 115 \\ 58 & 98 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 32A7 | $32$ | $16002$ | $C_{64}$ | 16A7 | 32A11 | 32A15 | 32A7 | $ \left[ \left(\begin{array}{rr} 70 & 52 \\ 45 & 28 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 32A9 | $32$ | $16002$ | $C_{64}$ | 16A7 | 32A5 | 32A1 | 32A9 | $ \left[ \left(\begin{array}{rr} 68 & 60 \\ 91 & 0 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 32A11 | $32$ | $16002$ | $C_{64}$ | 16A5 | 32A1 | 32A13 | 32A11 | $ \left[ \left(\begin{array}{rr} 73 & 119 \\ 81 & 99 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 32A13 | $32$ | $16002$ | $C_{64}$ | 16A3 | 32A7 | 32A5 | 32A13 | $ \left[ \left(\begin{array}{rr} 6 & 32 \\ 57 & 29 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 32A15 | $32$ | $16002$ | $C_{64}$ | 16A1 | 32A13 | 32A9 | 32A15 | $ \left[ \left(\begin{array}{rr} 75 & 120 \\ 55 & 66 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A1 | $63$ | $16256$ | $C_{63}$ | 63A2 | 21A1 | 9A1 | 63A1 | $ \left[ \left(\begin{array}{rr} 0 & 11 \\ 23 & 104 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A2 | $63$ | $16256$ | $C_{63}$ | 63A4 | 21A2 | 9A2 | 63A2 | $ \left[ \left(\begin{array}{rr} 126 & 1 \\ 106 & 20 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A4 | $63$ | $16256$ | $C_{63}$ | 63A8 | 21A4 | 9A4 | 63A4 | $ \left[ \left(\begin{array}{rr} 107 & 19 \\ 109 & 125 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A5 | $63$ | $16256$ | $C_{63}$ | 63A10 | 21A5 | 9A4 | 63A5 | $ \left[ \left(\begin{array}{rr} 56 & 105 \\ 81 & 102 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A8 | $63$ | $16256$ | $C_{63}$ | 63A16 | 21A8 | 9A1 | 63A8 | $ \left[ \left(\begin{array}{rr} 58 & 90 \\ 15 & 43 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A10 | $63$ | $16256$ | $C_{63}$ | 63A20 | 21A10 | 9A1 | 63A10 | $ \left[ \left(\begin{array}{rr} 43 & 47 \\ 29 & 14 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A11 | $63$ | $16256$ | $C_{63}$ | 63A22 | 21A10 | 9A2 | 63A11 | $ \left[ \left(\begin{array}{rr} 65 & 27 \\ 68 & 124 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A13 | $63$ | $16256$ | $C_{63}$ | 63A26 | 21A8 | 9A4 | 63A13 | $ \left[ \left(\begin{array}{rr} 3 & 97 \\ 122 & 8 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A16 | $63$ | $16256$ | $C_{63}$ | 63A31 | 21A5 | 9A2 | 63A16 | $ \left[ \left(\begin{array}{rr} 15 & 73 \\ 118 & 24 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A17 | $63$ | $16256$ | $C_{63}$ | 63A29 | 21A4 | 9A1 | 63A17 | $ \left[ \left(\begin{array}{rr} 28 & 10 \\ 44 & 111 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A19 | $63$ | $16256$ | $C_{63}$ | 63A25 | 21A2 | 9A1 | 63A19 | $ \left[ \left(\begin{array}{rr} 111 & 26 \\ 89 & 22 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A20 | $63$ | $16256$ | $C_{63}$ | 63A23 | 21A1 | 9A2 | 63A20 | $ \left[ \left(\begin{array}{rr} 37 & 12 \\ 2 & 35 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A22 | $63$ | $16256$ | $C_{63}$ | 63A19 | 21A1 | 9A4 | 63A22 | $ \left[ \left(\begin{array}{rr} 35 & 104 \\ 102 & 60 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A23 | $63$ | $16256$ | $C_{63}$ | 63A17 | 21A2 | 9A4 | 63A23 | $ \left[ \left(\begin{array}{rr} 106 & 25 \\ 110 & 123 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A25 | $63$ | $16256$ | $C_{63}$ | 63A13 | 21A4 | 9A2 | 63A25 | $ \left[ \left(\begin{array}{rr} 4 & 98 \\ 101 & 30 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A26 | $63$ | $16256$ | $C_{63}$ | 63A11 | 21A5 | 9A1 | 63A26 | $ \left[ \left(\begin{array}{rr} 95 & 76 \\ 55 & 40 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A29 | $63$ | $16256$ | $C_{63}$ | 63A5 | 21A8 | 9A2 | 63A29 | $ \left[ \left(\begin{array}{rr} 66 & 120 \\ 20 & 46 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 63A31 | $63$ | $16256$ | $C_{63}$ | 63A1 | 21A10 | 9A4 | 63A31 | $ \left[ \left(\begin{array}{rr} 81 & 53 \\ 30 & 51 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 64A1 | $64$ | $16002$ | $C_{64}$ | 32A1 | 64A3 | 64A7 | 64A1 | $ \left[ \left(\begin{array}{rr} 85 & 124 \\ 78 & 63 \end{array}\right) \right] $ |
| $\PSL(2,127)$ | 64A3 | $64$ | $16002$ | $C_{64}$ | 32A3 | 64A9 | 64A21 | 64A3 | $ \left[ \left(\begin{array}{rr} 86 & 50 \\ 97 & 114 \end{array}\right) \right] $ |