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Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\GL(2,149)$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | ||||
|---|---|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 5P | 37P | 149P | ||||||
| $\PGL(2,149)$ | 1A | $1$ | $1$ | $\PGL(2,149)$ | 1A | 1A | 1A | 1A | 1A | $ \left[ \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 2A | $2$ | $11026$ | $D_{150}$ | 1A | 2A | 2A | 2A | 2A | $ \left[ \left(\begin{array}{rr} 136 & 60 \\ 4 & 13 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 2B | $2$ | $11175$ | $D_{148}$ | 1A | 2B | 2B | 2B | 2B | $ \left[ \left(\begin{array}{rr} 58 & 0 \\ 0 & 91 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 3A | $3$ | $22052$ | $C_{150}$ | 3A | 1A | 3A | 3A | 3A | $ \left[ \left(\begin{array}{rr} 22 & 103 \\ 136 & 12 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 4A | $4$ | $22350$ | $C_{148}$ | 2B | 4A | 4A | 4A | 4A | $ \left[ \left(\begin{array}{rr} 41 & 0 \\ 0 & 133 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 5A1 | $5$ | $22052$ | $C_{150}$ | 5A2 | 5A2 | 1A | 5A2 | 5A1 | $ \left[ \left(\begin{array}{rr} 37 & 25 \\ 101 & 23 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 5A2 | $5$ | $22052$ | $C_{150}$ | 5A1 | 5A1 | 1A | 5A1 | 5A2 | $ \left[ \left(\begin{array}{rr} 4 & 2 \\ 20 & 134 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 6A | $6$ | $22052$ | $C_{150}$ | 3A | 2A | 6A | 6A | 6A | $ \left[ \left(\begin{array}{rr} 59 & 42 \\ 122 & 107 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 10A1 | $10$ | $22052$ | $C_{150}$ | 5A1 | 10A3 | 2A | 10A3 | 10A1 | $ \left[ \left(\begin{array}{rr} 17 & 119 \\ 147 & 4 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 10A3 | $10$ | $22052$ | $C_{150}$ | 5A2 | 10A1 | 2A | 10A1 | 10A3 | $ \left[ \left(\begin{array}{rr} 9 & 114 \\ 97 & 118 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 15A1 | $15$ | $22052$ | $C_{150}$ | 15A2 | 5A1 | 3A | 15A7 | 15A1 | $ \left[ \left(\begin{array}{rr} 29 & 37 \\ 72 & 50 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 15A2 | $15$ | $22052$ | $C_{150}$ | 15A4 | 5A2 | 3A | 15A1 | 15A2 | $ \left[ \left(\begin{array}{rr} 123 & 19 \\ 41 & 17 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 15A4 | $15$ | $22052$ | $C_{150}$ | 15A7 | 5A1 | 3A | 15A2 | 15A4 | $ \left[ \left(\begin{array}{rr} 105 & 66 \\ 64 & 74 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 15A7 | $15$ | $22052$ | $C_{150}$ | 15A1 | 5A2 | 3A | 15A4 | 15A7 | $ \left[ \left(\begin{array}{rr} 26 & 144 \\ 99 & 148 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 25A1 | $25$ | $22052$ | $C_{150}$ | 25A2 | 25A3 | 5A1 | 25A12 | 25A1 | $ \left[ \left(\begin{array}{rr} 63 & 118 \\ 137 & 134 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 25A2 | $25$ | $22052$ | $C_{150}$ | 25A4 | 25A6 | 5A2 | 25A1 | 25A2 | $ \left[ \left(\begin{array}{rr} 63 & 51 \\ 63 & 100 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 25A3 | $25$ | $22052$ | $C_{150}$ | 25A6 | 25A9 | 5A2 | 25A11 | 25A3 | $ \left[ \left(\begin{array}{rr} 46 & 79 \\ 45 & 115 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 25A4 | $25$ | $22052$ | $C_{150}$ | 25A8 | 25A12 | 5A1 | 25A2 | 25A4 | $ \left[ \left(\begin{array}{rr} 40 & 58 \\ 133 & 85 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 25A6 | $25$ | $22052$ | $C_{150}$ | 25A12 | 25A7 | 5A1 | 25A3 | 25A6 | $ \left[ \left(\begin{array}{rr} 114 & 88 \\ 135 & 23 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 25A7 | $25$ | $22052$ | $C_{150}$ | 25A11 | 25A4 | 5A2 | 25A9 | 25A7 | $ \left[ \left(\begin{array}{rr} 42 & 68 \\ 84 & 141 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 25A8 | $25$ | $22052$ | $C_{150}$ | 25A9 | 25A1 | 5A2 | 25A4 | 25A8 | $ \left[ \left(\begin{array}{rr} 147 & 68 \\ 84 & 97 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 25A9 | $25$ | $22052$ | $C_{150}$ | 25A7 | 25A2 | 5A1 | 25A8 | 25A9 | $ \left[ \left(\begin{array}{rr} 66 & 88 \\ 135 & 124 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 25A11 | $25$ | $22052$ | $C_{150}$ | 25A3 | 25A8 | 5A1 | 25A7 | 25A11 | $ \left[ \left(\begin{array}{rr} 2 & 64 \\ 44 & 139 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 25A12 | $25$ | $22052$ | $C_{150}$ | 25A1 | 25A11 | 5A2 | 25A6 | 25A12 | $ \left[ \left(\begin{array}{rr} 23 & 64 \\ 44 & 11 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 30A1 | $30$ | $22052$ | $C_{150}$ | 15A1 | 10A1 | 6A | 30A7 | 30A1 | $ \left[ \left(\begin{array}{rr} 104 & 88 \\ 135 & 13 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 30A7 | $30$ | $22052$ | $C_{150}$ | 15A7 | 10A3 | 6A | 30A11 | 30A7 | $ \left[ \left(\begin{array}{rr} 89 & 107 \\ 27 & 41 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 30A11 | $30$ | $22052$ | $C_{150}$ | 15A4 | 10A1 | 6A | 30A13 | 30A11 | $ \left[ \left(\begin{array}{rr} 146 & 65 \\ 54 & 50 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 30A13 | $30$ | $22052$ | $C_{150}$ | 15A2 | 10A3 | 6A | 30A1 | 30A13 | $ \left[ \left(\begin{array}{rr} 91 & 114 \\ 97 & 51 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A1 | $37$ | $22350$ | $C_{148}$ | 37A2 | 37A3 | 37A5 | 1A | 37A1 | $ \left[ \left(\begin{array}{rr} 38 & 0 \\ 0 & 74 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A2 | $37$ | $22350$ | $C_{148}$ | 37A4 | 37A6 | 37A10 | 1A | 37A2 | $ \left[ \left(\begin{array}{rr} 66 & 0 \\ 0 & 79 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A3 | $37$ | $22350$ | $C_{148}$ | 37A6 | 37A9 | 37A15 | 1A | 37A3 | $ \left[ \left(\begin{array}{rr} 113 & 0 \\ 0 & 110 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A4 | $37$ | $22350$ | $C_{148}$ | 37A8 | 37A12 | 37A17 | 1A | 37A4 | $ \left[ \left(\begin{array}{rr} 29 & 0 \\ 0 & 37 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A5 | $37$ | $22350$ | $C_{148}$ | 37A10 | 37A15 | 37A12 | 1A | 37A5 | $ \left[ \left(\begin{array}{rr} 63 & 0 \\ 0 & 80 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A6 | $37$ | $22350$ | $C_{148}$ | 37A12 | 37A18 | 37A7 | 1A | 37A6 | $ \left[ \left(\begin{array}{rr} 101 & 0 \\ 0 & 43 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A7 | $37$ | $22350$ | $C_{148}$ | 37A14 | 37A16 | 37A2 | 1A | 37A7 | $ \left[ \left(\begin{array}{rr} 83 & 0 \\ 0 & 122 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A8 | $37$ | $22350$ | $C_{148}$ | 37A16 | 37A13 | 37A3 | 1A | 37A8 | $ \left[ \left(\begin{array}{rr} 5 & 0 \\ 0 & 107 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A9 | $37$ | $22350$ | $C_{148}$ | 37A18 | 37A10 | 37A8 | 1A | 37A9 | $ \left[ \left(\begin{array}{rr} 135 & 0 \\ 0 & 11 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A10 | $37$ | $22350$ | $C_{148}$ | 37A17 | 37A7 | 37A13 | 1A | 37A10 | $ \left[ \left(\begin{array}{rr} 61 & 0 \\ 0 & 110 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A11 | $37$ | $22350$ | $C_{148}$ | 37A15 | 37A4 | 37A18 | 1A | 37A11 | $ \left[ \left(\begin{array}{rr} 14 & 0 \\ 0 & 111 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A12 | $37$ | $22350$ | $C_{148}$ | 37A13 | 37A1 | 37A14 | 1A | 37A12 | $ \left[ \left(\begin{array}{rr} 35 & 0 \\ 0 & 113 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A13 | $37$ | $22350$ | $C_{148}$ | 37A11 | 37A2 | 37A9 | 1A | 37A13 | $ \left[ \left(\begin{array}{rr} 85 & 0 \\ 0 & 37 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A14 | $37$ | $22350$ | $C_{148}$ | 37A9 | 37A5 | 37A4 | 1A | 37A14 | $ \left[ \left(\begin{array}{rr} 75 & 0 \\ 0 & 136 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A15 | $37$ | $22350$ | $C_{148}$ | 37A7 | 37A8 | 37A1 | 1A | 37A15 | $ \left[ \left(\begin{array}{rr} 75 & 0 \\ 0 & 108 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A16 | $37$ | $22350$ | $C_{148}$ | 37A5 | 37A11 | 37A6 | 1A | 37A16 | $ \left[ \left(\begin{array}{rr} 44 & 0 \\ 0 & 71 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A17 | $37$ | $22350$ | $C_{148}$ | 37A3 | 37A14 | 37A11 | 1A | 37A17 | $ \left[ \left(\begin{array}{rr} 100 & 0 \\ 0 & 64 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 37A18 | $37$ | $22350$ | $C_{148}$ | 37A1 | 37A17 | 37A16 | 1A | 37A18 | $ \left[ \left(\begin{array}{rr} 42 & 0 \\ 0 & 143 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 50A1 | $50$ | $22052$ | $C_{150}$ | 25A1 | 50A3 | 10A1 | 50A13 | 50A1 | $ \left[ \left(\begin{array}{rr} 147 & 32 \\ 22 & 141 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 50A3 | $50$ | $22052$ | $C_{150}$ | 25A3 | 50A9 | 10A3 | 50A11 | 50A3 | $ \left[ \left(\begin{array}{rr} 105 & 51 \\ 63 & 142 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 50A7 | $50$ | $22052$ | $C_{150}$ | 25A7 | 50A21 | 10A3 | 50A9 | 50A7 | $ \left[ \left(\begin{array}{rr} 71 & 27 \\ 121 & 38 \end{array}\right) \right] $ |
| $\PGL(2,149)$ | 50A9 | $50$ | $22052$ | $C_{150}$ | 25A9 | 50A23 | 10A1 | 50A17 | 50A9 | $ \left[ \left(\begin{array}{rr} 87 & 17 \\ 21 & 0 \end{array}\right) \right] $ |