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Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\Sp(4,7)$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |||
|---|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 5P | 7P | ||||||
| $\PSp(4,7)$ | 1A | $1$ | $1$ | not computed | 1A | 1A | 1A | 1A | $ \left[ \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 2A | $2$ | $1225$ | not computed | 1A | 2A | 2A | 2A | $ \left[ \left(\begin{array}{rrrr} 0 & 6 & 0 & 0 \\ 6 & 0 & 0 & 0 \\ 1 & 0 & 0 & 6 \\ 0 & 6 & 6 & 0 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 2B | $2$ | $51450$ | not computed | 1A | 2B | 2B | 2B | $ \left[ \left(\begin{array}{rrrr} 4 & 4 & 2 & 6 \\ 3 & 5 & 5 & 2 \\ 3 & 1 & 2 & 3 \\ 0 & 3 & 4 & 3 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 3A | $3$ | $137200$ | not computed | 3A | 1A | 3A | 3A | $ \left[ \left(\begin{array}{rrrr} 3 & 1 & 0 & 5 \\ 3 & 0 & 1 & 0 \\ 1 & 3 & 1 & 6 \\ 5 & 1 & 4 & 5 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 3B | $3$ | $137200$ | not computed | 3B | 1A | 3B | 3B | $ \left[ \left(\begin{array}{rrrr} 6 & 4 & 4 & 1 \\ 2 & 0 & 3 & 3 \\ 3 & 0 & 2 & 5 \\ 2 & 6 & 1 & 5 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 4A | $4$ | $102900$ | not computed | 2B | 4A | 4A | 4A | $ \left[ \left(\begin{array}{rrrr} 2 & 5 & 5 & 1 \\ 1 & 1 & 2 & 5 \\ 5 & 0 & 3 & 2 \\ 1 & 5 & 6 & 2 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 4B | $4$ | $102900$ | not computed | 2A | 4B | 4B | 4B | $ \left[ \left(\begin{array}{rrrr} 6 & 0 & 3 & 5 \\ 4 & 2 & 2 & 2 \\ 6 & 1 & 0 & 2 \\ 1 & 6 & 6 & 4 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 4C | $4$ | $2160900$ | not computed | 2B | 4C | 4C | 4C | $ \left[ \left(\begin{array}{rrrr} 4 & 4 & 1 & 0 \\ 1 & 0 & 2 & 5 \\ 4 & 1 & 0 & 2 \\ 4 & 2 & 3 & 3 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 5A | $5$ | $5531904$ | not computed | 5A | 5A | 1A | 5A | $ \left[ \left(\begin{array}{rrrr} 1 & 6 & 5 & 3 \\ 1 & 2 & 4 & 3 \\ 6 & 0 & 5 & 2 \\ 3 & 2 & 6 & 5 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 6A | $6$ | $137200$ | not computed | 3B | 2A | 6A | 6A | $ \left[ \left(\begin{array}{rrrr} 4 & 1 & 6 & 1 \\ 1 & 1 & 3 & 1 \\ 4 & 0 & 3 & 2 \\ 2 & 5 & 0 & 3 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 6B | $6$ | $2881200$ | not computed | 3A | 2B | 6B | 6B | $ \left[ \left(\begin{array}{rrrr} 6 & 3 & 2 & 0 \\ 5 & 2 & 0 & 3 \\ 4 & 6 & 1 & 5 \\ 4 & 3 & 4 & 5 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 6C | $6$ | $3841600$ | not computed | 3A | 2A | 6C | 6C | $ \left[ \left(\begin{array}{rrrr} 2 & 2 & 6 & 6 \\ 1 & 0 & 1 & 2 \\ 5 & 4 & 5 & 1 \\ 3 & 1 & 2 & 0 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 7A1 | $7$ | $1200$ | not computed | 7A1 | 7A-1 | 7A-1 | 1A | $ \left[ \left(\begin{array}{rrrr} 5 & 4 & 1 & 6 \\ 1 & 2 & 6 & 1 \\ 3 & 2 & 3 & 3 \\ 1 & 3 & 6 & 0 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 7A-1 | $7$ | $1200$ | not computed | 7A-1 | 7A1 | 7A1 | 1A | $ \left[ \left(\begin{array}{rrrr} 0 & 3 & 6 & 1 \\ 6 & 3 & 1 & 6 \\ 4 & 5 & 2 & 4 \\ 6 & 4 & 1 & 5 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 7B | $7$ | $50400$ | not computed | 7B | 7B | 7B | 1A | $ \left[ \left(\begin{array}{rrrr} 2 & 1 & 4 & 3 \\ 1 & 1 & 6 & 4 \\ 6 & 4 & 1 & 6 \\ 3 & 6 & 6 & 0 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 7C | $7$ | $67200$ | not computed | 7C | 7C | 7C | 1A | $ \left[ \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 6 & 6 & 2 & 0 \\ 6 & 5 & 3 & 0 \\ 4 & 6 & 1 & 1 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 7D1 | $7$ | $2822400$ | not computed | 7D1 | 7D-1 | 7D-1 | 1A | $ \left[ \left(\begin{array}{rrrr} 4 & 2 & 3 & 1 \\ 3 & 3 & 5 & 4 \\ 5 & 5 & 2 & 4 \\ 6 & 3 & 4 & 2 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 7D-1 | $7$ | $2822400$ | not computed | 7D-1 | 7D1 | 7D1 | 1A | $ \left[ \left(\begin{array}{rrrr} 5 & 3 & 4 & 1 \\ 3 & 5 & 5 & 3 \\ 3 & 5 & 4 & 5 \\ 6 & 5 & 4 & 3 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 8A1 | $8$ | $102900$ | not computed | 4B | 8A3 | 8A3 | 8A1 | $ \left[ \left(\begin{array}{rrrr} 6 & 2 & 0 & 3 \\ 4 & 4 & 4 & 3 \\ 5 & 2 & 0 & 6 \\ 2 & 5 & 1 & 2 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 8A3 | $8$ | $102900$ | not computed | 4B | 8A1 | 8A1 | 8A3 | $ \left[ \left(\begin{array}{rrrr} 5 & 1 & 5 & 4 \\ 4 & 2 & 3 & 6 \\ 2 & 5 & 6 & 4 \\ 5 & 2 & 0 & 2 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 8B1 | $8$ | $2881200$ | not computed | 4A | 8B3 | 8B3 | 8B1 | $ \left[ \left(\begin{array}{rrrr} 1 & 3 & 2 & 4 \\ 4 & 2 & 5 & 5 \\ 3 & 1 & 0 & 0 \\ 4 & 0 & 2 & 4 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 8B3 | $8$ | $2881200$ | not computed | 4A | 8B1 | 8B1 | 8B3 | $ \left[ \left(\begin{array}{rrrr} 0 & 5 & 6 & 6 \\ 0 & 1 & 4 & 1 \\ 5 & 5 & 5 & 4 \\ 6 & 3 & 4 & 1 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 8C | $8$ | $4321800$ | not computed | 4B | 8C | 8C | 8C | $ \left[ \left(\begin{array}{rrrr} 3 & 1 & 2 & 1 \\ 4 & 1 & 1 & 4 \\ 2 & 6 & 4 & 1 \\ 4 & 1 & 5 & 3 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 12A | $12$ | $5762400$ | not computed | 6B | 4A | 12A | 12A | $ \left[ \left(\begin{array}{rrrr} 4 & 1 & 1 & 4 \\ 2 & 2 & 5 & 3 \\ 6 & 3 & 2 & 1 \\ 3 & 4 & 2 & 2 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 12B | $12$ | $5762400$ | not computed | 6A | 4B | 12B | 12B | $ \left[ \left(\begin{array}{rrrr} 3 & 4 & 1 & 1 \\ 5 & 2 & 6 & 6 \\ 6 & 4 & 6 & 0 \\ 4 & 5 & 1 & 2 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 14A1 | $14$ | $58800$ | not computed | 7A1 | 14A-1 | 14A-1 | 2A | $ \left[ \left(\begin{array}{rrrr} 0 & 3 & 2 & 4 \\ 0 & 6 & 4 & 4 \\ 2 & 6 & 2 & 0 \\ 3 & 2 & 1 & 6 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 14A-1 | $14$ | $58800$ | not computed | 7A-1 | 14A1 | 14A1 | 2A | $ \left[ \left(\begin{array}{rrrr} 6 & 0 & 3 & 3 \\ 1 & 2 & 3 & 5 \\ 5 & 1 & 6 & 3 \\ 4 & 5 & 0 & 0 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 14B1 | $14$ | $705600$ | not computed | 7B | 14B-1 | 14B-1 | 2A | $ \left[ \left(\begin{array}{rrrr} 0 & 2 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 5 & 0 & 6 & 0 \\ 1 & 0 & 1 & 0 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 14B-1 | $14$ | $705600$ | not computed | 7B | 14B1 | 14B1 | 2A | $ \left[ \left(\begin{array}{rrrr} 0 & 0 & 1 & 1 \\ 6 & 1 & 1 & 1 \\ 0 & 0 & 6 & 5 \\ 1 & 5 & 6 & 0 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 14C | $14$ | $1411200$ | not computed | 7C | 14C | 14C | 2A | $ \left[ \left(\begin{array}{rrrr} 6 & 4 & 2 & 1 \\ 2 & 2 & 5 & 0 \\ 5 & 4 & 4 & 3 \\ 3 & 0 & 5 & 2 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 14D | $14$ | $2469600$ | not computed | 7B | 14D | 14D | 2B | $ \left[ \left(\begin{array}{rrrr} 6 & 2 & 1 & 4 \\ 2 & 4 & 1 & 4 \\ 5 & 0 & 5 & 4 \\ 4 & 0 & 1 & 6 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 21A1 | $21$ | $3292800$ | not computed | 21A1 | 7A-1 | 21A-1 | 3B | $ \left[ \left(\begin{array}{rrrr} 2 & 6 & 5 & 6 \\ 2 & 5 & 4 & 3 \\ 1 & 4 & 6 & 2 \\ 0 & 2 & 3 & 0 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 21A-1 | $21$ | $3292800$ | not computed | 21A-1 | 7A1 | 21A1 | 3B | $ \left[ \left(\begin{array}{rrrr} 0 & 5 & 3 & 6 \\ 4 & 1 & 4 & 5 \\ 2 & 4 & 2 & 1 \\ 0 & 1 & 5 & 5 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 21B | $21$ | $6585600$ | not computed | 21B | 7C | 21B | 3A | $ \left[ \left(\begin{array}{rrrr} 5 & 4 & 3 & 0 \\ 3 & 1 & 4 & 3 \\ 2 & 0 & 5 & 3 \\ 3 & 0 & 6 & 1 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 24A1 | $24$ | $5762400$ | not computed | 12A | 8B1 | 24A5 | 24A1 | $ \left[ \left(\begin{array}{rrrr} 5 & 2 & 6 & 3 \\ 2 & 2 & 2 & 0 \\ 6 & 2 & 0 & 6 \\ 5 & 5 & 0 & 5 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 24A5 | $24$ | $5762400$ | not computed | 12A | 8B3 | 24A1 | 24A5 | $ \left[ \left(\begin{array}{rrrr} 6 & 1 & 0 & 2 \\ 2 & 1 & 6 & 2 \\ 0 & 5 & 1 & 1 \\ 5 & 5 & 2 & 5 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 24B1 | $24$ | $5762400$ | not computed | 12B | 8A1 | 24B5 | 24B1 | $ \left[ \left(\begin{array}{rrrr} 3 & 2 & 5 & 5 \\ 3 & 1 & 0 & 2 \\ 6 & 1 & 1 & 4 \\ 0 & 1 & 6 & 5 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 24B5 | $24$ | $5762400$ | not computed | 12B | 8A3 | 24B1 | 24B5 | $ \left[ \left(\begin{array}{rrrr} 2 & 2 & 4 & 2 \\ 3 & 0 & 0 & 3 \\ 2 & 6 & 0 & 0 \\ 0 & 5 & 0 & 0 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 25A1 | $25$ | $5531904$ | not computed | 25A2 | 25A3 | 5A | 25A1 | $ \left[ \left(\begin{array}{rrrr} 2 & 5 & 6 & 2 \\ 5 & 3 & 4 & 4 \\ 6 & 5 & 3 & 3 \\ 5 & 2 & 2 & 3 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 25A2 | $25$ | $5531904$ | not computed | 25A3 | 25A6 | 5A | 25A2 | $ \left[ \left(\begin{array}{rrrr} 5 & 3 & 5 & 6 \\ 6 & 6 & 6 & 4 \\ 0 & 3 & 1 & 1 \\ 5 & 5 & 1 & 5 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 25A3 | $25$ | $5531904$ | not computed | 25A6 | 25A9 | 5A | 25A3 | $ \left[ \left(\begin{array}{rrrr} 1 & 1 & 6 & 6 \\ 0 & 2 & 2 & 3 \\ 5 & 2 & 3 & 4 \\ 3 & 6 & 0 & 6 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 25A6 | $25$ | $5531904$ | not computed | 25A9 | 25A1 | 5A | 25A6 | $ \left[ \left(\begin{array}{rrrr} 0 & 5 & 2 & 1 \\ 2 & 2 & 4 & 3 \\ 3 & 3 & 6 & 5 \\ 0 & 5 & 5 & 5 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 25A9 | $25$ | $5531904$ | not computed | 25A1 | 25A2 | 5A | 25A9 | $ \left[ \left(\begin{array}{rrrr} 6 & 6 & 2 & 1 \\ 3 & 4 & 0 & 3 \\ 6 & 2 & 0 & 4 \\ 5 & 1 & 4 & 2 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 28A1 | $28$ | $2469600$ | not computed | 14A1 | 28A-1 | 28A-1 | 4B | $ \left[ \left(\begin{array}{rrrr} 5 & 4 & 0 & 0 \\ 6 & 1 & 4 & 1 \\ 0 & 4 & 4 & 1 \\ 6 & 0 & 5 & 6 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 28A-1 | $28$ | $2469600$ | not computed | 14A-1 | 28A1 | 28A1 | 4B | $ \left[ \left(\begin{array}{rrrr} 1 & 6 & 1 & 0 \\ 2 & 3 & 4 & 0 \\ 0 & 4 & 6 & 3 \\ 6 & 0 & 1 & 2 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 28B | $28$ | $4939200$ | not computed | 14D | 28B | 28B | 4A | $ \left[ \left(\begin{array}{rrrr} 0 & 3 & 2 & 4 \\ 5 & 2 & 1 & 6 \\ 6 & 2 & 0 & 5 \\ 4 & 4 & 6 & 6 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 42A1 | $42$ | $3292800$ | not computed | 21A1 | 14A-1 | 42A-1 | 6A | $ \left[ \left(\begin{array}{rrrr} 4 & 5 & 0 & 0 \\ 2 & 6 & 6 & 3 \\ 1 & 4 & 1 & 4 \\ 0 & 3 & 1 & 6 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 42A-1 | $42$ | $3292800$ | not computed | 21A-1 | 14A1 | 42A1 | 6A | $ \left[ \left(\begin{array}{rrrr} 6 & 4 & 4 & 0 \\ 1 & 1 & 1 & 0 \\ 4 & 3 & 6 & 5 \\ 0 & 6 & 2 & 4 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 56A1 | $56$ | $2469600$ | not computed | 28A1 | 56A3 | 56A3 | 8A1 | $ \left[ \left(\begin{array}{rrrr} 4 & 5 & 4 & 2 \\ 0 & 3 & 3 & 1 \\ 1 & 4 & 0 & 1 \\ 3 & 1 & 2 & 2 \end{array}\right) \right] $ |
| $\PSp(4,7)$ | 56A-1 | $56$ | $2469600$ | not computed | 28A-1 | 56A-3 | 56A-3 | 8A1 | $ \left[ \left(\begin{array}{rrrr} 2 & 1 & 6 & 5 \\ 2 & 0 & 4 & 3 \\ 6 & 3 & 3 & 5 \\ 4 & 6 & 0 & 4 \end{array}\right) \right] $ |