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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 | ||||||
| $\He_7:(C_6^2:C_2^2)$ | 1A | $1$ | $1$ | $\He_7:(C_6^2:C_2^2)$ | 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) $ |
| $\He_7:(C_6^2:C_2^2)$ | 2A | $2$ | $1$ | $\He_7:(C_6^2:C_2^2)$ | 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) $ |
| $\He_7:(C_6^2:C_2^2)$ | 2B | $2$ | $49$ | $C_2\times D_{42}:C_6$ | 1A | 2B | 2B | $ \left(\begin{array}{rrrr} 5 & 1 & 5 & 0 \\ 5 & 2 & 0 & 2 \\ 4 & 0 & 2 & 1 \\ 0 & 3 & 5 & 5 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 2C | $2$ | $49$ | $C_2\times D_{42}:C_6$ | 1A | 2C | 2C | $ \left(\begin{array}{rrrr} 5 & 4 & 2 & 0 \\ 3 & 2 & 0 & 5 \\ 3 & 0 & 2 & 4 \\ 0 & 4 & 3 & 5 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 2D | $2$ | $98$ | $C_{14}:C_6^2$ | 1A | 2D | 2D | $ \left(\begin{array}{rrrr} 0 & 0 & 6 & 0 \\ 1 & 0 & 3 & 6 \\ 6 & 0 & 0 & 0 \\ 4 & 6 & 6 & 0 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 2E | $2$ | $98$ | $C_{14}:C_6^2$ | 1A | 2E | 2E | $ \left(\begin{array}{rrrr} 4 & 4 & 6 & 3 \\ 4 & 2 & 0 & 6 \\ 5 & 4 & 5 & 3 \\ 3 & 5 & 3 & 3 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 2F | $2$ | $294$ | $C_2^2\times F_7$ | 1A | 2F | 2F | $ \left(\begin{array}{rrrr} 5 & 0 & 4 & 3 \\ 0 & 5 & 0 & 4 \\ 1 & 1 & 2 & 0 \\ 0 & 1 & 0 & 2 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 2G | $2$ | $294$ | $C_2^2\times F_7$ | 1A | 2G | 2G | $ \left(\begin{array}{rrrr} 1 & 1 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 1 & 6 \\ 0 & 0 & 0 & 6 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 3A | $3$ | $98$ | $C_{14}:C_6^2$ | 3A | 1A | 3A | $ \left(\begin{array}{rrrr} 1 & 0 & 1 & 3 \\ 4 & 0 & 5 & 5 \\ 4 & 6 & 1 & 4 \\ 5 & 4 & 0 & 6 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 3B1 | $3$ | $98$ | $C_{14}:C_6^2$ | 3B-1 | 1A | 3B1 | $ \left(\begin{array}{rrrr} 3 & 5 & 5 & 3 \\ 5 & 2 & 2 & 5 \\ 1 & 6 & 4 & 2 \\ 2 & 1 & 2 & 3 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 3B-1 | $3$ | $98$ | $C_{14}:C_6^2$ | 3B1 | 1A | 3B-1 | $ \left(\begin{array}{rrrr} 3 & 2 & 2 & 4 \\ 2 & 4 & 5 & 2 \\ 6 & 1 & 2 & 5 \\ 5 & 6 & 5 & 3 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 3C1 | $3$ | $343$ | $C_6^2:C_2^2$ | 3C-1 | 1A | 3C1 | $ \left(\begin{array}{rrrr} 5 & 5 & 5 & 1 \\ 6 & 5 & 4 & 5 \\ 3 & 1 & 5 & 5 \\ 0 & 6 & 2 & 0 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 3C-1 | $3$ | $343$ | $C_6^2:C_2^2$ | 3C1 | 1A | 3C-1 | $ \left(\begin{array}{rrrr} 0 & 5 & 2 & 6 \\ 2 & 5 & 3 & 2 \\ 1 & 6 & 5 & 5 \\ 0 & 4 & 6 & 5 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 4A | $4$ | $294$ | $C_{14}:C_{12}$ | 2B | 4A | 4A | $ \left(\begin{array}{rrrr} 3 & 3 & 0 & 2 \\ 5 & 6 & 4 & 4 \\ 0 & 2 & 2 & 5 \\ 2 & 1 & 6 & 5 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 4B | $4$ | $294$ | $C_{14}:C_{12}$ | 2B | 4B | 4B | $ \left(\begin{array}{rrrr} 5 & 6 & 4 & 1 \\ 3 & 2 & 5 & 6 \\ 6 & 6 & 0 & 6 \\ 6 & 6 & 1 & 5 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6A | $6$ | $98$ | $C_{14}:C_6^2$ | 3A | 2A | 6A | $ \left(\begin{array}{rrrr} 1 & 3 & 5 & 3 \\ 0 & 6 & 5 & 1 \\ 4 & 6 & 0 & 0 \\ 5 & 4 & 3 & 6 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6B | $6$ | $98$ | $C_{14}:C_6^2$ | 3A | 2B | 6B | $ \left(\begin{array}{rrrr} 4 & 6 & 1 & 4 \\ 1 & 0 & 6 & 6 \\ 5 & 1 & 1 & 2 \\ 3 & 6 & 1 & 5 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6C | $6$ | $98$ | $C_{14}:C_6^2$ | 3A | 2C | 6C | $ \left(\begin{array}{rrrr} 0 & 6 & 6 & 6 \\ 1 & 3 & 4 & 5 \\ 6 & 3 & 2 & 2 \\ 1 & 1 & 1 & 6 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6D1 | $6$ | $98$ | $C_{14}:C_6^2$ | 3B-1 | 2A | 6D1 | $ \left(\begin{array}{rrrr} 1 & 5 & 2 & 5 \\ 4 & 4 & 1 & 2 \\ 4 & 1 & 4 & 2 \\ 4 & 4 & 3 & 0 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6D-1 | $6$ | $98$ | $C_{14}:C_6^2$ | 3B1 | 2A | 6D-1 | $ \left(\begin{array}{rrrr} 0 & 2 & 5 & 2 \\ 3 & 4 & 6 & 5 \\ 3 & 6 & 4 & 5 \\ 3 & 3 & 4 & 1 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6E1 | $6$ | $98$ | $C_{14}:C_6^2$ | 3B1 | 2D | 6E1 | $ \left(\begin{array}{rrrr} 4 & 4 & 2 & 0 \\ 0 & 1 & 2 & 2 \\ 0 & 1 & 1 & 3 \\ 0 & 0 & 0 & 5 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6E-1 | $6$ | $98$ | $C_{14}:C_6^2$ | 3B-1 | 2D | 6E-1 | $ \left(\begin{array}{rrrr} 2 & 4 & 2 & 0 \\ 0 & 6 & 2 & 2 \\ 0 & 1 & 6 & 3 \\ 0 & 0 & 0 & 3 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6F1 | $6$ | $98$ | $C_{14}:C_6^2$ | 3B-1 | 2E | 6F1 | $ \left(\begin{array}{rrrr} 0 & 6 & 6 & 5 \\ 2 & 5 & 2 & 6 \\ 4 & 3 & 4 & 1 \\ 0 & 4 & 5 & 2 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6F-1 | $6$ | $98$ | $C_{14}:C_6^2$ | 3B1 | 2E | 6F-1 | $ \left(\begin{array}{rrrr} 5 & 6 & 6 & 5 \\ 2 & 3 & 2 & 6 \\ 4 & 3 & 2 & 1 \\ 0 & 4 & 5 & 0 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6G1 | $6$ | $343$ | $C_6^2:C_2^2$ | 3C1 | 2A | 6G1 | $ \left(\begin{array}{rrrr} 5 & 0 & 4 & 4 \\ 3 & 3 & 0 & 2 \\ 1 & 1 & 5 & 3 \\ 5 & 0 & 1 & 0 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6G-1 | $6$ | $343$ | $C_6^2:C_2^2$ | 3C-1 | 2A | 6G-1 | $ \left(\begin{array}{rrrr} 0 & 3 & 5 & 3 \\ 1 & 5 & 0 & 3 \\ 0 & 6 & 3 & 0 \\ 2 & 6 & 3 & 5 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6H1 | $6$ | $343$ | $C_6^2:C_2^2$ | 3C1 | 2C | 6H1 | $ \left(\begin{array}{rrrr} 0 & 1 & 1 & 2 \\ 1 & 2 & 4 & 0 \\ 1 & 2 & 4 & 1 \\ 0 & 3 & 4 & 4 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6H-1 | $6$ | $343$ | $C_6^2:C_2^2$ | 3C-1 | 2C | 6H-1 | $ \left(\begin{array}{rrrr} 4 & 1 & 0 & 5 \\ 4 & 4 & 3 & 6 \\ 4 & 5 & 2 & 1 \\ 0 & 6 & 1 & 0 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6I1 | $6$ | $343$ | $C_6^2:C_2^2$ | 3C-1 | 2B | 6I1 | $ \left(\begin{array}{rrrr} 5 & 2 & 2 & 3 \\ 3 & 6 & 4 & 4 \\ 1 & 3 & 3 & 2 \\ 0 & 5 & 6 & 4 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6I-1 | $6$ | $343$ | $C_6^2:C_2^2$ | 3C1 | 2B | 6I-1 | $ \left(\begin{array}{rrrr} 4 & 2 & 3 & 4 \\ 6 & 3 & 3 & 5 \\ 2 & 4 & 6 & 2 \\ 0 & 6 & 3 & 5 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6J1 | $6$ | $686$ | $C_2\times C_6^2$ | 3B1 | 2B | 6J1 | $ \left(\begin{array}{rrrr} 5 & 5 & 4 & 5 \\ 1 & 2 & 2 & 2 \\ 1 & 2 & 3 & 1 \\ 5 & 5 & 1 & 4 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6J-1 | $6$ | $686$ | $C_2\times C_6^2$ | 3B-1 | 2B | 6J-1 | $ \left(\begin{array}{rrrr} 4 & 1 & 5 & 2 \\ 1 & 3 & 5 & 3 \\ 2 & 5 & 2 & 5 \\ 2 & 6 & 1 & 5 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6K1 | $6$ | $686$ | $C_2\times C_6^2$ | 3C1 | 2D | 6K1 | $ \left(\begin{array}{rrrr} 6 & 1 & 2 & 2 \\ 3 & 5 & 4 & 2 \\ 0 & 2 & 5 & 1 \\ 4 & 2 & 0 & 3 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6K-1 | $6$ | $686$ | $C_2\times C_6^2$ | 3C-1 | 2D | 6K-1 | $ \left(\begin{array}{rrrr} 4 & 6 & 2 & 2 \\ 0 & 2 & 4 & 2 \\ 2 & 2 & 2 & 6 \\ 4 & 0 & 4 & 1 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6L1 | $6$ | $686$ | $C_2\times C_6^2$ | 3A | 2D | 6L1 | $ \left(\begin{array}{rrrr} 6 & 0 & 2 & 6 \\ 6 & 0 & 5 & 6 \\ 4 & 5 & 3 & 1 \\ 3 & 2 & 6 & 0 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6L-1 | $6$ | $686$ | $C_2\times C_6^2$ | 3A | 2D | 6L-1 | $ \left(\begin{array}{rrrr} 0 & 6 & 6 & 6 \\ 1 & 4 & 5 & 2 \\ 2 & 5 & 0 & 0 \\ 3 & 4 & 1 & 1 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6M1 | $6$ | $686$ | $C_2\times C_6^2$ | 3B-1 | 2C | 6M1 | $ \left(\begin{array}{rrrr} 5 & 1 & 3 & 2 \\ 6 & 3 & 4 & 2 \\ 0 & 3 & 2 & 0 \\ 6 & 6 & 4 & 4 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6M-1 | $6$ | $686$ | $C_2\times C_6^2$ | 3B1 | 2C | 6M-1 | $ \left(\begin{array}{rrrr} 4 & 0 & 5 & 5 \\ 4 & 2 & 3 & 4 \\ 1 & 4 & 3 & 1 \\ 1 & 0 & 6 & 5 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6N1 | $6$ | $686$ | $C_2\times C_6^2$ | 3B1 | 2D | 6N1 | $ \left(\begin{array}{rrrr} 0 & 4 & 0 & 3 \\ 6 & 4 & 1 & 6 \\ 5 & 4 & 2 & 3 \\ 1 & 2 & 4 & 1 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6N-1 | $6$ | $686$ | $C_2\times C_6^2$ | 3B-1 | 2D | 6N-1 | $ \left(\begin{array}{rrrr} 6 & 4 & 6 & 3 \\ 3 & 5 & 1 & 0 \\ 2 & 4 & 3 & 3 \\ 1 & 5 & 1 & 0 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6O1 | $6$ | $686$ | $C_2\times C_6^2$ | 3A | 2E | 6O1 | $ \left(\begin{array}{rrrr} 2 & 4 & 3 & 2 \\ 2 & 0 & 5 & 6 \\ 1 & 6 & 5 & 0 \\ 3 & 0 & 2 & 2 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6O-1 | $6$ | $686$ | $C_2\times C_6^2$ | 3A | 2E | 6O-1 | $ \left(\begin{array}{rrrr} 5 & 0 & 6 & 2 \\ 5 & 2 & 5 & 3 \\ 0 & 6 & 0 & 3 \\ 3 & 1 & 5 & 5 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6P1 | $6$ | $686$ | $C_2\times C_6^2$ | 3C1 | 2E | 6P1 | $ \left(\begin{array}{rrrr} 5 & 4 & 5 & 1 \\ 4 & 3 & 4 & 2 \\ 0 & 1 & 5 & 4 \\ 4 & 1 & 0 & 3 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6P-1 | $6$ | $686$ | $C_2\times C_6^2$ | 3C-1 | 2E | 6P-1 | $ \left(\begin{array}{rrrr} 4 & 3 & 2 & 1 \\ 0 & 2 & 4 & 5 \\ 1 & 1 & 4 & 3 \\ 4 & 0 & 3 & 2 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6Q1 | $6$ | $686$ | $C_2\times C_6^2$ | 3B-1 | 2E | 6Q1 | $ \left(\begin{array}{rrrr} 0 & 4 & 4 & 4 \\ 1 & 1 & 2 & 6 \\ 2 & 3 & 5 & 5 \\ 2 & 3 & 3 & 1 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6Q-1 | $6$ | $686$ | $C_2\times C_6^2$ | 3B1 | 2E | 6Q-1 | $ \left(\begin{array}{rrrr} 6 & 2 & 6 & 4 \\ 4 & 2 & 2 & 4 \\ 3 & 3 & 6 & 3 \\ 2 & 2 & 6 & 0 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6R1 | $6$ | $2058$ | $C_2^2\times C_6$ | 3C1 | 2F | 6R1 | $ \left(\begin{array}{rrrr} 5 & 6 & 2 & 1 \\ 4 & 2 & 1 & 6 \\ 6 & 6 & 5 & 6 \\ 2 & 0 & 5 & 4 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6R-1 | $6$ | $2058$ | $C_2^2\times C_6$ | 3C-1 | 2F | 6R-1 | $ \left(\begin{array}{rrrr} 3 & 1 & 6 & 1 \\ 2 & 2 & 1 & 2 \\ 0 & 6 & 5 & 1 \\ 2 & 6 & 3 & 2 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6S1 | $6$ | $2058$ | $C_2^2\times C_6$ | 3C1 | 2G | 6S1 | $ \left(\begin{array}{rrrr} 0 & 0 & 5 & 6 \\ 3 & 2 & 1 & 4 \\ 2 & 6 & 1 & 6 \\ 3 & 5 & 2 & 2 \end{array}\right) $ |
| $\He_7:(C_6^2:C_2^2)$ | 6S-1 | $6$ | $2058$ | $C_2^2\times C_6$ | 3C-1 | 2G | 6S-1 | $ \left(\begin{array}{rrrr} 5 & 1 & 4 & 6 \\ 5 & 6 & 1 & 5 \\ 5 & 6 & 5 & 0 \\ 3 & 2 & 4 & 0 \end{array}\right) $ |