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Elements of the group are displayed as matrices in $\GL_{2}(\F_{337})$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | ||
|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 7P | ||||||
| $D_{56}:C_6$ | 1A | $1$ | $1$ | $D_{56}:C_6$ | 1A | 1A | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right) $ |
| $D_{56}:C_6$ | 2A | $2$ | $1$ | $D_{56}:C_6$ | 1A | 2A | 2A | $ \left(\begin{array}{rr} 336 & 0 \\ 0 & 336 \end{array}\right) $ |
| $D_{56}:C_6$ | 2B | $2$ | $2$ | $C_2\times C_{168}$ | 1A | 2B | 2B | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 336 \end{array}\right) $ |
| $D_{56}:C_6$ | 2C | $2$ | $28$ | $C_2\times C_{12}$ | 1A | 2C | 2C | $ \left(\begin{array}{rr} 0 & 173 \\ 150 & 0 \end{array}\right) $ |
| $D_{56}:C_6$ | 2D | $2$ | $28$ | $C_2\times C_{12}$ | 1A | 2D | 2D | $ \left(\begin{array}{rr} 0 & 7 \\ 289 & 0 \end{array}\right) $ |
| $D_{56}:C_6$ | 3A1 | $3$ | $1$ | $D_{56}:C_6$ | 3A-1 | 1A | 3A1 | $ \left(\begin{array}{rr} 208 & 0 \\ 0 & 208 \end{array}\right) $ |
| $D_{56}:C_6$ | 3A-1 | $3$ | $1$ | $D_{56}:C_6$ | 3A1 | 1A | 3A-1 | $ \left(\begin{array}{rr} 128 & 0 \\ 0 & 128 \end{array}\right) $ |
| $D_{56}:C_6$ | 4A1 | $4$ | $1$ | $D_{56}:C_6$ | 2A | 4A-1 | 4A-1 | $ \left(\begin{array}{rr} 148 & 0 \\ 0 & 148 \end{array}\right) $ |
| $D_{56}:C_6$ | 4A-1 | $4$ | $1$ | $D_{56}:C_6$ | 2A | 4A1 | 4A1 | $ \left(\begin{array}{rr} 189 & 0 \\ 0 & 189 \end{array}\right) $ |
| $D_{56}:C_6$ | 4B | $4$ | $2$ | $C_2\times C_{168}$ | 2A | 4B | 4B | $ \left(\begin{array}{rr} 148 & 0 \\ 0 & 189 \end{array}\right) $ |
| $D_{56}:C_6$ | 4C | $4$ | $28$ | $C_2\times C_{12}$ | 2A | 4C | 4C | $ \left(\begin{array}{rr} 0 & 295 \\ 329 & 0 \end{array}\right) $ |
| $D_{56}:C_6$ | 4D | $4$ | $28$ | $C_2\times C_{12}$ | 2A | 4D | 4D | $ \left(\begin{array}{rr} 0 & 25 \\ 310 & 0 \end{array}\right) $ |
| $D_{56}:C_6$ | 6A1 | $6$ | $1$ | $D_{56}:C_6$ | 3A-1 | 2A | 6A1 | $ \left(\begin{array}{rr} 129 & 0 \\ 0 & 129 \end{array}\right) $ |
| $D_{56}:C_6$ | 6A-1 | $6$ | $1$ | $D_{56}:C_6$ | 3A1 | 2A | 6A-1 | $ \left(\begin{array}{rr} 209 & 0 \\ 0 & 209 \end{array}\right) $ |
| $D_{56}:C_6$ | 6B1 | $6$ | $2$ | $C_2\times C_{168}$ | 3A1 | 2B | 6B1 | $ \left(\begin{array}{rr} 128 & 0 \\ 0 & 209 \end{array}\right) $ |
| $D_{56}:C_6$ | 6B-1 | $6$ | $2$ | $C_2\times C_{168}$ | 3A-1 | 2B | 6B-1 | $ \left(\begin{array}{rr} 208 & 0 \\ 0 & 129 \end{array}\right) $ |
| $D_{56}:C_6$ | 6C1 | $6$ | $28$ | $C_2\times C_{12}$ | 3A1 | 2C | 6C1 | $ \left(\begin{array}{rr} 0 & 239 \\ 328 & 0 \end{array}\right) $ |
| $D_{56}:C_6$ | 6C-1 | $6$ | $28$ | $C_2\times C_{12}$ | 3A-1 | 2C | 6C-1 | $ \left(\begin{array}{rr} 0 & 262 \\ 196 & 0 \end{array}\right) $ |
| $D_{56}:C_6$ | 6D1 | $6$ | $28$ | $C_2\times C_{12}$ | 3A-1 | 2D | 6D1 | $ \left(\begin{array}{rr} 0 & 108 \\ 126 & 0 \end{array}\right) $ |
| $D_{56}:C_6$ | 6D-1 | $6$ | $28$ | $C_2\times C_{12}$ | 3A1 | 2D | 6D-1 | $ \left(\begin{array}{rr} 0 & 222 \\ 259 & 0 \end{array}\right) $ |
| $D_{56}:C_6$ | 7A1 | $7$ | $2$ | $C_2\times C_{168}$ | 7A2 | 7A3 | 1A | $ \left(\begin{array}{rr} 64 & 0 \\ 0 & 79 \end{array}\right) $ |
| $D_{56}:C_6$ | 7A2 | $7$ | $2$ | $C_2\times C_{168}$ | 7A3 | 7A1 | 1A | $ \left(\begin{array}{rr} 52 & 0 \\ 0 & 175 \end{array}\right) $ |
| $D_{56}:C_6$ | 7A3 | $7$ | $2$ | $C_2\times C_{168}$ | 7A1 | 7A2 | 1A | $ \left(\begin{array}{rr} 295 & 0 \\ 0 & 8 \end{array}\right) $ |
| $D_{56}:C_6$ | 8A1 | $8$ | $2$ | $C_2\times C_{168}$ | 4B | 8A3 | 8A1 | $ \left(\begin{array}{rr} 252 & 0 \\ 0 & 111 \end{array}\right) $ |
| $D_{56}:C_6$ | 8A3 | $8$ | $2$ | $C_2\times C_{168}$ | 4B | 8A1 | 8A3 | $ \left(\begin{array}{rr} 226 & 0 \\ 0 & 85 \end{array}\right) $ |
| $D_{56}:C_6$ | 8B1 | $8$ | $2$ | $C_2\times C_{168}$ | 4B | 8B1 | 8B-1 | $ \left(\begin{array}{rr} 252 & 0 \\ 0 & 226 \end{array}\right) $ |
| $D_{56}:C_6$ | 8B-1 | $8$ | $2$ | $C_2\times C_{168}$ | 4B | 8B-1 | 8B1 | $ \left(\begin{array}{rr} 111 & 0 \\ 0 & 85 \end{array}\right) $ |
| $D_{56}:C_6$ | 12A1 | $12$ | $1$ | $D_{56}:C_6$ | 6A1 | 4A1 | 12A-5 | $ \left(\begin{array}{rr} 265 & 0 \\ 0 & 265 \end{array}\right) $ |
| $D_{56}:C_6$ | 12A-1 | $12$ | $1$ | $D_{56}:C_6$ | 6A-1 | 4A-1 | 12A5 | $ \left(\begin{array}{rr} 117 & 0 \\ 0 & 117 \end{array}\right) $ |
| $D_{56}:C_6$ | 12A5 | $12$ | $1$ | $D_{56}:C_6$ | 6A-1 | 4A1 | 12A-1 | $ \left(\begin{array}{rr} 220 & 0 \\ 0 & 220 \end{array}\right) $ |
| $D_{56}:C_6$ | 12A-5 | $12$ | $1$ | $D_{56}:C_6$ | 6A1 | 4A-1 | 12A1 | $ \left(\begin{array}{rr} 72 & 0 \\ 0 & 72 \end{array}\right) $ |
| $D_{56}:C_6$ | 12B1 | $12$ | $2$ | $C_2\times C_{168}$ | 6A1 | 4B | 12B1 | $ \left(\begin{array}{rr} 265 & 0 \\ 0 & 72 \end{array}\right) $ |
| $D_{56}:C_6$ | 12B-1 | $12$ | $2$ | $C_2\times C_{168}$ | 6A-1 | 4B | 12B-1 | $ \left(\begin{array}{rr} 117 & 0 \\ 0 & 220 \end{array}\right) $ |
| $D_{56}:C_6$ | 12C1 | $12$ | $28$ | $C_2\times C_{12}$ | 6A-1 | 4C | 12C1 | $ \left(\begin{array}{rr} 0 & 311 \\ 316 & 0 \end{array}\right) $ |
| $D_{56}:C_6$ | 12C-1 | $12$ | $28$ | $C_2\times C_{12}$ | 6A1 | 4C | 12C-1 | $ \left(\begin{array}{rr} 0 & 16 \\ 324 & 0 \end{array}\right) $ |
| $D_{56}:C_6$ | 12D1 | $12$ | $28$ | $C_2\times C_{12}$ | 6A-1 | 4D | 12D1 | $ \left(\begin{array}{rr} 0 & 192 \\ 224 & 0 \end{array}\right) $ |
| $D_{56}:C_6$ | 12D-1 | $12$ | $28$ | $C_2\times C_{12}$ | 6A1 | 4D | 12D-1 | $ \left(\begin{array}{rr} 0 & 167 \\ 251 & 0 \end{array}\right) $ |
| $D_{56}:C_6$ | 14A1 | $14$ | $2$ | $C_2\times C_{168}$ | 7A1 | 14A3 | 2A | $ \left(\begin{array}{rr} 329 & 0 \\ 0 & 42 \end{array}\right) $ |
| $D_{56}:C_6$ | 14A3 | $14$ | $2$ | $C_2\times C_{168}$ | 7A3 | 14A5 | 2A | $ \left(\begin{array}{rr} 162 & 0 \\ 0 & 285 \end{array}\right) $ |
| $D_{56}:C_6$ | 14A5 | $14$ | $2$ | $C_2\times C_{168}$ | 7A2 | 14A1 | 2A | $ \left(\begin{array}{rr} 258 & 0 \\ 0 & 273 \end{array}\right) $ |
| $D_{56}:C_6$ | 14B1 | $14$ | $2$ | $C_2\times C_{168}$ | 7A1 | 14B3 | 2B | $ \left(\begin{array}{rr} 8 & 0 \\ 0 & 42 \end{array}\right) $ |
| $D_{56}:C_6$ | 14B-1 | $14$ | $2$ | $C_2\times C_{168}$ | 7A1 | 14B-3 | 2B | $ \left(\begin{array}{rr} 295 & 0 \\ 0 & 329 \end{array}\right) $ |
| $D_{56}:C_6$ | 14B3 | $14$ | $2$ | $C_2\times C_{168}$ | 7A3 | 14B-5 | 2B | $ \left(\begin{array}{rr} 175 & 0 \\ 0 & 285 \end{array}\right) $ |
| $D_{56}:C_6$ | 14B-3 | $14$ | $2$ | $C_2\times C_{168}$ | 7A3 | 14B5 | 2B | $ \left(\begin{array}{rr} 52 & 0 \\ 0 & 162 \end{array}\right) $ |
| $D_{56}:C_6$ | 14B5 | $14$ | $2$ | $C_2\times C_{168}$ | 7A2 | 14B1 | 2B | $ \left(\begin{array}{rr} 79 & 0 \\ 0 & 273 \end{array}\right) $ |
| $D_{56}:C_6$ | 14B-5 | $14$ | $2$ | $C_2\times C_{168}$ | 7A2 | 14B-1 | 2B | $ \left(\begin{array}{rr} 64 & 0 \\ 0 & 258 \end{array}\right) $ |
| $D_{56}:C_6$ | 21A1 | $21$ | $2$ | $C_2\times C_{168}$ | 21A2 | 7A1 | 3A1 | $ \left(\begin{array}{rr} 4 & 0 \\ 0 & 32 \end{array}\right) $ |
| $D_{56}:C_6$ | 21A-1 | $21$ | $2$ | $C_2\times C_{168}$ | 21A-2 | 7A1 | 3A-1 | $ \left(\begin{array}{rr} 253 & 0 \\ 0 & 158 \end{array}\right) $ |
| $D_{56}:C_6$ | 21A2 | $21$ | $2$ | $C_2\times C_{168}$ | 21A4 | 7A2 | 3A-1 | $ \left(\begin{array}{rr} 16 & 0 \\ 0 & 13 \end{array}\right) $ |
| $D_{56}:C_6$ | 21A-2 | $21$ | $2$ | $C_2\times C_{168}$ | 21A-4 | 7A2 | 3A1 | $ \left(\begin{array}{rr} 316 & 0 \\ 0 & 26 \end{array}\right) $ |