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Elements of the group are displayed as matrices in $\GL_{4}(\F_{7})$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | ||
|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 7P | ||||||
| $C_3\times \SL(2,7):C_2^2$ | 1A | $1$ | $1$ | $C_3\times \SL(2,7):C_2^2$ | 1A | 1A | 1A | $ \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 2A | $2$ | $1$ | $C_3\times \SL(2,7):C_2^2$ | 1A | 2A | 2A | $ \left(\begin{array}{rrrr} 6 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 2B | $2$ | $42$ | $C_6\times D_8$ | 1A | 2B | 2B | $ \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 5 & 0 & 6 & 0 \\ 0 & 2 & 0 & 1 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 2C | $2$ | $42$ | $C_6\times D_8$ | 1A | 2C | 2C | $ \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 4 & 6 & 4 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 3 & 6 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 2D | $2$ | $42$ | $C_6\times D_8$ | 1A | 2D | 2D | $ \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 3 & 1 & 3 & 0 \\ 5 & 0 & 6 & 0 \\ 2 & 5 & 4 & 6 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 3A1 | $3$ | $1$ | $C_3\times \SL(2,7):C_2^2$ | 3A-1 | 1A | 3A1 | $ \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 3A-1 | $3$ | $1$ | $C_3\times \SL(2,7):C_2^2$ | 3A1 | 1A | 3A-1 | $ \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 3B | $3$ | $56$ | $Q_8\times C_3^2$ | 3B | 1A | 3B | $ \left(\begin{array}{rrrr} 4 & 2 & 4 & 4 \\ 2 & 4 & 4 & 4 \\ 4 & 2 & 2 & 5 \\ 2 & 4 & 5 & 2 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 3C1 | $3$ | $56$ | $Q_8\times C_3^2$ | 3C-1 | 1A | 3C1 | $ \left(\begin{array}{rrrr} 6 & 6 & 0 & 1 \\ 6 & 6 & 1 & 0 \\ 2 & 0 & 4 & 1 \\ 0 & 2 & 1 & 4 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 3C-1 | $3$ | $56$ | $Q_8\times C_3^2$ | 3C1 | 1A | 3C-1 | $ \left(\begin{array}{rrrr} 2 & 4 & 0 & 3 \\ 4 & 2 & 3 & 0 \\ 6 & 0 & 3 & 3 \\ 0 & 6 & 3 & 3 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 4A | $4$ | $2$ | $\SL(2,7):C_6$ | 2A | 4A | 4A | $ \left(\begin{array}{rrrr} 2 & 2 & 0 & 2 \\ 5 & 5 & 5 & 0 \\ 0 & 4 & 2 & 5 \\ 3 & 0 & 2 & 5 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 4B | $4$ | $2$ | $\SL(2,7):C_6$ | 2A | 4B | 4B | $ \left(\begin{array}{rrrr} 3 & 0 & 3 & 0 \\ 1 & 4 & 0 & 4 \\ 6 & 0 & 4 & 0 \\ 0 & 1 & 1 & 3 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 4C | $4$ | $2$ | $\SL(2,7):C_6$ | 2A | 4C | 4C | $ \left(\begin{array}{rrrr} 1 & 3 & 1 & 0 \\ 1 & 6 & 0 & 6 \\ 2 & 0 & 6 & 3 \\ 0 & 5 & 1 & 1 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 4D | $4$ | $42$ | $\OD_{16}:C_6$ | 2A | 4D | 4D | $ \left(\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 5 & 6 & 0 \\ 5 & 0 & 0 & 6 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 6A1 | $6$ | $1$ | $C_3\times \SL(2,7):C_2^2$ | 3A-1 | 2A | 6A1 | $ \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 6A-1 | $6$ | $1$ | $C_3\times \SL(2,7):C_2^2$ | 3A1 | 2A | 6A-1 | $ \left(\begin{array}{rrrr} 5 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 5 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 6B1 | $6$ | $42$ | $C_6\times D_8$ | 3A1 | 2B | 6B1 | $ \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 \\ 3 & 0 & 5 & 0 \\ 0 & 4 & 0 & 2 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 6B-1 | $6$ | $42$ | $C_6\times D_8$ | 3A-1 | 2B | 6B-1 | $ \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 1 & 0 & 4 & 0 \\ 0 & 6 & 0 & 3 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 6C1 | $6$ | $42$ | $C_6\times D_8$ | 3A1 | 2C | 6C1 | $ \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 1 & 5 & 1 & 0 \\ 0 & 0 & 2 & 0 \\ 2 & 0 & 6 & 5 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 6C-1 | $6$ | $42$ | $C_6\times D_8$ | 3A-1 | 2C | 6C-1 | $ \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 5 & 4 & 5 & 0 \\ 0 & 0 & 3 & 0 \\ 3 & 0 & 2 & 4 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 6D1 | $6$ | $42$ | $C_6\times D_8$ | 3A1 | 2D | 6D1 | $ \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 6 & 2 & 6 & 0 \\ 3 & 0 & 5 & 0 \\ 4 & 3 & 1 & 5 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 6D-1 | $6$ | $42$ | $C_6\times D_8$ | 3A-1 | 2D | 6D-1 | $ \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 2 & 3 & 2 & 0 \\ 1 & 0 & 4 & 0 \\ 6 & 1 & 5 & 4 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 6E | $6$ | $56$ | $Q_8\times C_3^2$ | 3B | 2A | 6E | $ \left(\begin{array}{rrrr} 5 & 2 & 4 & 4 \\ 2 & 5 & 4 & 4 \\ 4 & 2 & 3 & 5 \\ 2 & 4 & 5 & 3 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 6F1 | $6$ | $56$ | $Q_8\times C_3^2$ | 3C1 | 2A | 6F1 | $ \left(\begin{array}{rrrr} 5 & 3 & 0 & 4 \\ 3 & 5 & 4 & 0 \\ 1 & 0 & 4 & 4 \\ 0 & 1 & 4 & 4 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 6F-1 | $6$ | $56$ | $Q_8\times C_3^2$ | 3C-1 | 2A | 6F-1 | $ \left(\begin{array}{rrrr} 1 & 1 & 0 & 6 \\ 1 & 1 & 6 & 0 \\ 5 & 0 & 3 & 6 \\ 0 & 5 & 6 & 3 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 7A1 | $7$ | $24$ | $Q_8\times C_{21}$ | 7A1 | 7A-1 | 1A | $ \left(\begin{array}{rrrr} 1 & 0 & 1 & 6 \\ 0 & 1 & 6 & 1 \\ 2 & 2 & 1 & 0 \\ 2 & 2 & 0 & 1 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 7A-1 | $7$ | $24$ | $Q_8\times C_{21}$ | 7A-1 | 7A1 | 1A | $ \left(\begin{array}{rrrr} 1 & 0 & 6 & 1 \\ 0 & 1 & 1 & 6 \\ 5 & 5 & 1 & 0 \\ 5 & 5 & 0 & 1 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 8A1 | $8$ | $42$ | $\OD_{16}:C_6$ | 4D | 8A3 | 8A1 | $ \left(\begin{array}{rrrr} 4 & 0 & 0 & 2 \\ 0 & 4 & 2 & 0 \\ 0 & 3 & 0 & 0 \\ 3 & 0 & 0 & 0 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 8A3 | $8$ | $42$ | $\OD_{16}:C_6$ | 4D | 8A1 | 8A3 | $ \left(\begin{array}{rrrr} 0 & 0 & 0 & 2 \\ 0 & 0 & 2 & 0 \\ 0 & 3 & 3 & 0 \\ 3 & 0 & 0 & 3 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 8B | $8$ | $84$ | $C_2\times C_{24}$ | 4D | 8B | 8B | $ \left(\begin{array}{rrrr} 6 & 4 & 5 & 5 \\ 0 & 6 & 6 & 1 \\ 0 & 3 & 4 & 1 \\ 2 & 6 & 4 & 5 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 8C | $8$ | $84$ | $C_2\times C_{24}$ | 4D | 8C | 8C | $ \left(\begin{array}{rrrr} 6 & 6 & 0 & 2 \\ 5 & 3 & 6 & 5 \\ 2 & 5 & 2 & 1 \\ 0 & 5 & 5 & 3 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 8D | $8$ | $84$ | $C_2\times C_{24}$ | 4D | 8D | 8D | $ \left(\begin{array}{rrrr} 1 & 5 & 6 & 3 \\ 2 & 6 & 4 & 1 \\ 3 & 6 & 5 & 6 \\ 1 & 4 & 1 & 2 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12A1 | $12$ | $2$ | $\SL(2,7):C_6$ | 6A-1 | 4A | 12A1 | $ \left(\begin{array}{rrrr} 6 & 6 & 0 & 6 \\ 1 & 1 & 1 & 0 \\ 0 & 5 & 6 & 1 \\ 2 & 0 & 6 & 1 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12A-1 | $12$ | $2$ | $\SL(2,7):C_6$ | 6A1 | 4A | 12A-1 | $ \left(\begin{array}{rrrr} 4 & 4 & 0 & 4 \\ 3 & 3 & 3 & 0 \\ 0 & 1 & 4 & 3 \\ 6 & 0 & 4 & 3 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12B1 | $12$ | $2$ | $\SL(2,7):C_6$ | 6A-1 | 4B | 12B1 | $ \left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 3 & 5 & 0 & 5 \\ 4 & 0 & 5 & 0 \\ 0 & 3 & 3 & 2 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12B-1 | $12$ | $2$ | $\SL(2,7):C_6$ | 6A1 | 4B | 12B-1 | $ \left(\begin{array}{rrrr} 6 & 0 & 6 & 0 \\ 2 & 1 & 0 & 1 \\ 5 & 0 & 1 & 0 \\ 0 & 2 & 2 & 6 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12C1 | $12$ | $2$ | $\SL(2,7):C_6$ | 6A-1 | 4C | 12C1 | $ \left(\begin{array}{rrrr} 3 & 2 & 3 & 0 \\ 3 & 4 & 0 & 4 \\ 6 & 0 & 4 & 2 \\ 0 & 1 & 3 & 3 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12C-1 | $12$ | $2$ | $\SL(2,7):C_6$ | 6A1 | 4C | 12C-1 | $ \left(\begin{array}{rrrr} 2 & 6 & 2 & 0 \\ 2 & 5 & 0 & 5 \\ 4 & 0 & 5 & 6 \\ 0 & 3 & 2 & 2 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12D1 | $12$ | $42$ | $\OD_{16}:C_6$ | 6A1 | 4D | 12D1 | $ \left(\begin{array}{rrrr} 5 & 0 & 0 & 5 \\ 0 & 5 & 5 & 0 \\ 0 & 4 & 2 & 0 \\ 4 & 0 & 0 & 2 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12D-1 | $12$ | $42$ | $\OD_{16}:C_6$ | 6A-1 | 4D | 12D-1 | $ \left(\begin{array}{rrrr} 4 & 0 & 0 & 4 \\ 0 & 4 & 4 & 0 \\ 0 & 6 & 3 & 0 \\ 6 & 0 & 0 & 3 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12E | $12$ | $112$ | $C_3\times C_{12}$ | 6E | 4B | 12E | $ \left(\begin{array}{rrrr} 4 & 2 & 3 & 1 \\ 1 & 1 & 2 & 0 \\ 2 & 1 & 3 & 5 \\ 2 & 3 & 0 & 6 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12F | $12$ | $112$ | $C_3\times C_{12}$ | 6E | 4A | 12F | $ \left(\begin{array}{rrrr} 5 & 2 & 0 & 6 \\ 5 & 2 & 1 & 0 \\ 1 & 5 & 0 & 4 \\ 2 & 6 & 3 & 0 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12G | $12$ | $112$ | $C_3\times C_{12}$ | 6E | 4C | 12G | $ \left(\begin{array}{rrrr} 3 & 1 & 1 & 6 \\ 3 & 6 & 4 & 2 \\ 1 & 5 & 2 & 3 \\ 4 & 3 & 3 & 3 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12H1 | $12$ | $112$ | $C_3\times C_{12}$ | 6F1 | 4A | 12H1 | $ \left(\begin{array}{rrrr} 3 & 0 & 4 & 3 \\ 0 & 4 & 4 & 3 \\ 0 & 6 & 3 & 1 \\ 1 & 0 & 6 & 4 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12H-1 | $12$ | $112$ | $C_3\times C_{12}$ | 6F-1 | 4A | 12H-1 | $ \left(\begin{array}{rrrr} 2 & 4 & 2 & 2 \\ 3 & 5 & 5 & 5 \\ 0 & 4 & 2 & 0 \\ 3 & 0 & 0 & 5 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12I1 | $12$ | $112$ | $C_3\times C_{12}$ | 6F1 | 4C | 12I1 | $ \left(\begin{array}{rrrr} 6 & 4 & 1 & 5 \\ 4 & 6 & 4 & 4 \\ 1 & 3 & 4 & 1 \\ 5 & 0 & 0 & 5 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12I-1 | $12$ | $112$ | $C_3\times C_{12}$ | 6F-1 | 4C | 12I-1 | $ \left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 0 & 5 & 2 & 4 \\ 0 & 5 & 4 & 5 \\ 6 & 4 & 5 & 4 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12J1 | $12$ | $112$ | $C_3\times C_{12}$ | 6F1 | 4B | 12J1 | $ \left(\begin{array}{rrrr} 4 & 5 & 1 & 5 \\ 0 & 6 & 2 & 6 \\ 6 & 1 & 6 & 2 \\ 0 & 2 & 3 & 5 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 12J-1 | $12$ | $112$ | $C_3\times C_{12}$ | 6F-1 | 4B | 12J-1 | $ \left(\begin{array}{rrrr} 1 & 6 & 3 & 6 \\ 2 & 4 & 1 & 4 \\ 1 & 4 & 4 & 1 \\ 0 & 3 & 0 & 5 \end{array}\right) $ |
| $C_3\times \SL(2,7):C_2^2$ | 14A1 | $14$ | $24$ | $Q_8\times C_{21}$ | 7A1 | 14A-1 | 2A | $ \left(\begin{array}{rrrr} 6 & 0 & 3 & 4 \\ 0 & 6 & 4 & 3 \\ 6 & 6 & 6 & 0 \\ 6 & 6 & 0 & 6 \end{array}\right) $ |