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Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\SL(2,169)$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 5P | 7P | 13P | 17P | ||||||
| $\PSL(2,169)$ | 1A | $1$ | $1$ | $\PSL(2,169)$ | 1A | 1A | 1A | 1A | 1A | 1A | $ \left[ \left(\begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 2A | $2$ | $14365$ | $D_{84}$ | 1A | 2A | 2A | 2A | 2A | 2A | $ \left[ \left(\begin{array}{rr} 67 & 46 \\ 55 & 151 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 3A | $3$ | $28730$ | $C_{84}$ | 3A | 1A | 3A | 3A | 3A | 3A | $ \left[ \left(\begin{array}{rr} 68 & 144 \\ 153 & 63 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 4A | $4$ | $28730$ | $C_{84}$ | 2A | 4A | 4A | 4A | 4A | 4A | $ \left[ \left(\begin{array}{rr} 110 & 39 \\ 48 & 106 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 5A1 | $5$ | $28392$ | $C_{85}$ | 5A2 | 5A2 | 1A | 5A2 | 5A2 | 5A2 | $ \left[ \left(\begin{array}{rr} 22 & 12 \\ 34 & 79 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 5A2 | $5$ | $28392$ | $C_{85}$ | 5A1 | 5A1 | 1A | 5A1 | 5A1 | 5A1 | $ \left[ \left(\begin{array}{rr} 57 & 42 \\ 64 & 102 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 6A | $6$ | $28730$ | $C_{84}$ | 3A | 2A | 6A | 6A | 6A | 6A | $ \left[ \left(\begin{array}{rr} 134 & 116 \\ 125 & 16 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 7A1 | $7$ | $28730$ | $C_{84}$ | 7A2 | 7A3 | 7A2 | 1A | 7A1 | 7A3 | $ \left[ \left(\begin{array}{rr} 78 & 53 \\ 62 & 47 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 7A2 | $7$ | $28730$ | $C_{84}$ | 7A3 | 7A1 | 7A3 | 1A | 7A2 | 7A1 | $ \left[ \left(\begin{array}{rr} 146 & 109 \\ 118 & 15 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 7A3 | $7$ | $28730$ | $C_{84}$ | 7A1 | 7A2 | 7A1 | 1A | 7A3 | 7A2 | $ \left[ \left(\begin{array}{rr} 13 & 95 \\ 104 & 164 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 12A1 | $12$ | $28730$ | $C_{84}$ | 6A | 4A | 12A5 | 12A5 | 12A1 | 12A5 | $ \left[ \left(\begin{array}{rr} 13 & 151 \\ 160 & 45 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 12A5 | $12$ | $28730$ | $C_{84}$ | 6A | 4A | 12A1 | 12A1 | 12A5 | 12A1 | $ \left[ \left(\begin{array}{rr} 44 & 165 \\ 6 & 150 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 13A | $13$ | $14280$ | $C_{13}^2$ | 13A | 13A | 13A | 13A | 1A | 13A | $ \left[ \left(\begin{array}{rr} 145 & 124 \\ 76 & 59 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 13B | $13$ | $14280$ | $C_{13}^2$ | 13B | 13B | 13B | 13B | 1A | 13B | $ \left[ \left(\begin{array}{rr} 105 & 123 \\ 139 & 32 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 14A1 | $14$ | $28730$ | $C_{84}$ | 7A1 | 14A3 | 14A5 | 2A | 14A1 | 14A3 | $ \left[ \left(\begin{array}{rr} 5 & 74 \\ 83 & 66 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 14A3 | $14$ | $28730$ | $C_{84}$ | 7A3 | 14A5 | 14A1 | 2A | 14A3 | 14A5 | $ \left[ \left(\begin{array}{rr} 93 & 102 \\ 111 & 72 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 14A5 | $14$ | $28730$ | $C_{84}$ | 7A2 | 14A1 | 14A3 | 2A | 14A5 | 14A1 | $ \left[ \left(\begin{array}{rr} 14 & 4 \\ 13 & 116 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 17A1 | $17$ | $28392$ | $C_{85}$ | 17A2 | 17A3 | 17A5 | 17A7 | 17A4 | 1A | $ \left[ \left(\begin{array}{rr} 88 & 60 \\ 82 & 139 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 17A2 | $17$ | $28392$ | $C_{85}$ | 17A4 | 17A6 | 17A7 | 17A3 | 17A8 | 1A | $ \left[ \left(\begin{array}{rr} 109 & 139 \\ 161 & 62 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 17A3 | $17$ | $28392$ | $C_{85}$ | 17A6 | 17A8 | 17A2 | 17A4 | 17A5 | 1A | $ \left[ \left(\begin{array}{rr} 83 & 19 \\ 41 & 10 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 17A4 | $17$ | $28392$ | $C_{85}$ | 17A8 | 17A5 | 17A3 | 17A6 | 17A1 | 1A | $ \left[ \left(\begin{array}{rr} 87 & 78 \\ 100 & 97 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 17A5 | $17$ | $28392$ | $C_{85}$ | 17A7 | 17A2 | 17A8 | 17A1 | 17A3 | 1A | $ \left[ \left(\begin{array}{rr} 11 & 49 \\ 71 & 105 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 17A6 | $17$ | $28392$ | $C_{85}$ | 17A5 | 17A1 | 17A4 | 17A8 | 17A7 | 1A | $ \left[ \left(\begin{array}{rr} 51 & 11 \\ 33 & 39 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 17A7 | $17$ | $28392$ | $C_{85}$ | 17A3 | 17A4 | 17A1 | 17A2 | 17A6 | 1A | $ \left[ \left(\begin{array}{rr} 26 & 113 \\ 135 & 106 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 17A8 | $17$ | $28392$ | $C_{85}$ | 17A1 | 17A7 | 17A6 | 17A5 | 17A2 | 1A | $ \left[ \left(\begin{array}{rr} 159 & 97 \\ 119 & 5 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 21A1 | $21$ | $28730$ | $C_{84}$ | 21A2 | 7A1 | 21A5 | 3A | 21A8 | 21A4 | $ \left[ \left(\begin{array}{rr} 76 & 76 \\ 85 & 103 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 21A2 | $21$ | $28730$ | $C_{84}$ | 21A4 | 7A2 | 21A10 | 3A | 21A5 | 21A8 | $ \left[ \left(\begin{array}{rr} 128 & 147 \\ 156 & 158 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 21A4 | $21$ | $28730$ | $C_{84}$ | 21A8 | 7A3 | 21A1 | 3A | 21A10 | 21A5 | $ \left[ \left(\begin{array}{rr} 49 & 83 \\ 92 & 137 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 21A5 | $21$ | $28730$ | $C_{84}$ | 21A10 | 7A2 | 21A4 | 3A | 21A2 | 21A1 | $ \left[ \left(\begin{array}{rr} 64 & 99 \\ 108 & 106 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 21A8 | $21$ | $28730$ | $C_{84}$ | 21A5 | 7A1 | 21A2 | 3A | 21A1 | 21A10 | $ \left[ \left(\begin{array}{rr} 73 & 16 \\ 25 & 126 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 21A10 | $21$ | $28730$ | $C_{84}$ | 21A1 | 7A3 | 21A8 | 3A | 21A4 | 21A2 | $ \left[ \left(\begin{array}{rr} 155 & 23 \\ 32 & 60 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 28A1 | $28$ | $28730$ | $C_{84}$ | 14A1 | 28A3 | 28A5 | 4A | 28A13 | 28A11 | $ \left[ \left(\begin{array}{rr} 155 & 113 \\ 122 & 130 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 28A3 | $28$ | $28730$ | $C_{84}$ | 14A3 | 28A9 | 28A13 | 4A | 28A11 | 28A5 | $ \left[ \left(\begin{array}{rr} 31 & 73 \\ 82 & 42 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 28A5 | $28$ | $28730$ | $C_{84}$ | 14A5 | 28A13 | 28A3 | 4A | 28A9 | 28A1 | $ \left[ \left(\begin{array}{rr} 68 & 72 \\ 81 & 20 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 28A9 | $28$ | $28730$ | $C_{84}$ | 14A5 | 28A1 | 28A11 | 4A | 28A5 | 28A13 | $ \left[ \left(\begin{array}{rr} 133 & 48 \\ 57 & 16 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 28A11 | $28$ | $28730$ | $C_{84}$ | 14A3 | 28A5 | 28A1 | 4A | 28A3 | 28A9 | $ \left[ \left(\begin{array}{rr} 149 & 61 \\ 70 & 158 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 28A13 | $28$ | $28730$ | $C_{84}$ | 14A1 | 28A11 | 28A9 | 4A | 28A1 | 28A3 | $ \left[ \left(\begin{array}{rr} 76 & 77 \\ 86 & 24 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 42A1 | $42$ | $28730$ | $C_{84}$ | 21A1 | 14A1 | 42A5 | 6A | 42A13 | 42A17 | $ \left[ \left(\begin{array}{rr} 58 & 85 \\ 94 & 24 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 42A5 | $42$ | $28730$ | $C_{84}$ | 21A5 | 14A5 | 42A17 | 6A | 42A19 | 42A1 | $ \left[ \left(\begin{array}{rr} 149 & 31 \\ 40 & 91 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 42A11 | $42$ | $28730$ | $C_{84}$ | 21A10 | 14A3 | 42A13 | 6A | 42A17 | 42A19 | $ \left[ \left(\begin{array}{rr} 110 & 124 \\ 133 & 90 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 42A13 | $42$ | $28730$ | $C_{84}$ | 21A8 | 14A1 | 42A19 | 6A | 42A1 | 42A11 | $ \left[ \left(\begin{array}{rr} 61 & 133 \\ 142 & 20 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 42A17 | $42$ | $28730$ | $C_{84}$ | 21A4 | 14A3 | 42A1 | 6A | 42A11 | 42A5 | $ \left[ \left(\begin{array}{rr} 115 & 136 \\ 145 & 45 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 42A19 | $42$ | $28730$ | $C_{84}$ | 21A2 | 14A5 | 42A11 | 6A | 42A5 | 42A13 | $ \left[ \left(\begin{array}{rr} 83 & 19 \\ 28 & 130 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 84A1 | $84$ | $28730$ | $C_{84}$ | 42A1 | 28A1 | 84A5 | 12A1 | 84A13 | 84A17 | $ \left[ \left(\begin{array}{rr} 151 & 91 \\ 100 & 144 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 84A5 | $84$ | $28730$ | $C_{84}$ | 42A5 | 28A5 | 84A25 | 12A5 | 84A19 | 84A1 | $ \left[ \left(\begin{array}{rr} 83 & 9 \\ 18 & 80 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 84A11 | $84$ | $28730$ | $C_{84}$ | 42A11 | 28A11 | 84A29 | 12A1 | 84A25 | 84A19 | $ \left[ \left(\begin{array}{rr} 157 & 112 \\ 121 & 32 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 84A13 | $84$ | $28730$ | $C_{84}$ | 42A13 | 28A13 | 84A19 | 12A1 | 84A1 | 84A31 | $ \left[ \left(\begin{array}{rr} 14 & 127 \\ 136 & 63 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 84A17 | $84$ | $28730$ | $C_{84}$ | 42A17 | 28A11 | 84A1 | 12A5 | 84A31 | 84A37 | $ \left[ \left(\begin{array}{rr} 61 & 104 \\ 113 & 15 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 84A19 | $84$ | $28730$ | $C_{84}$ | 42A19 | 28A9 | 84A11 | 12A5 | 84A5 | 84A13 | $ \left[ \left(\begin{array}{rr} 62 & 153 \\ 162 & 6 \end{array}\right) \right] $ |
| $\PSL(2,169)$ | 84A23 | $84$ | $28730$ | $C_{84}$ | 42A19 | 28A5 | 84A31 | 12A1 | 84A37 | 84A29 | $ \left[ \left(\begin{array}{rr} 148 & 66 \\ 75 & 156 \end{array}\right) \right] $ |