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Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\SOPlus(4,9)$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | ||
|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 5P | ||||||
| $\PSOPlus(4,9)$ | 1A | $1$ | $1$ | $\PSOPlus(4,9)$ | 1A | 1A | 1A | $ \left[ \left(\begin{array}{rrrr} 0 & -1 & -1 & -1 \\ -1 & 0 & -1 & -1 \\ -1 & -1 & 0 & -1 \\ -1 & -1 & -1 & 0 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 2A | $2$ | $45$ | $D_4.A_6.C_2$ | 1A | 2A | 2A | $ \left[ \left(\begin{array}{rrrr} -1 & -1 & 2 & -1 \\ -1 & -1 & -1 & 6 \\ 2 & -1 & -1 & -1 \\ -1 & 6 & -1 & -1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 2B | $2$ | $45$ | $D_4.A_6.C_2$ | 1A | 2B | 2B | $ \left[ \left(\begin{array}{rrrr} 4 & 2 & -1 & -1 \\ 6 & 0 & -1 & -1 \\ -1 & -1 & 4 & 6 \\ -1 & -1 & 2 & 0 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 2C | $2$ | $1296$ | $D_5\times D_{10}$ | 1A | 2C | 2C | $ \left[ \left(\begin{array}{rrrr} 7 & 1 & 4 & 2 \\ 7 & 3 & 4 & 4 \\ 7 & 1 & 3 & 1 \\ 3 & 7 & 7 & 7 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 2D | $2$ | $2025$ | $D_4:D_8$ | 1A | 2D | 2D | $ \left[ \left(\begin{array}{rrrr} -1 & -1 & 5 & -1 \\ -1 & -1 & 1 & 5 \\ 3 & -1 & -1 & -1 \\ 3 & 3 & -1 & -1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 3A | $3$ | $80$ | $C_3^2\times A_6$ | 3A | 1A | 3A | $ \left[ \left(\begin{array}{rrrr} 1 & -1 & 3 & -1 \\ -1 & 1 & -1 & 7 \\ 7 & -1 & 6 & -1 \\ -1 & 3 & -1 & 6 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 3B | $3$ | $80$ | $C_3^2\times A_6$ | 3B | 1A | 3B | $ \left[ \left(\begin{array}{rrrr} 7 & 6 & -1 & -1 \\ 2 & 5 & -1 & -1 \\ -1 & -1 & 7 & 2 \\ -1 & -1 & 6 & 5 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 3C | $3$ | $3200$ | $C_3^4$ | 3C | 1A | 3C | $ \left[ \left(\begin{array}{rrrr} 4 & 1 & 4 & 5 \\ 5 & 6 & 5 & 2 \\ 4 & 1 & 3 & 4 \\ 1 & 2 & 0 & 5 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 3D | $3$ | $3200$ | $C_3^4$ | 3D | 1A | 3D | $ \left[ \left(\begin{array}{rrrr} -1 & 7 & -1 & -1 \\ 5 & 4 & -1 & -1 \\ -1 & 0 & -1 & 3 \\ 2 & 1 & 1 & 4 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 4A | $4$ | $90$ | $A_6:C_8$ | 2A | 4A | 4A | $ \left[ \left(\begin{array}{rrrr} 6 & -1 & 0 & -1 \\ -1 & 6 & -1 & 4 \\ 0 & -1 & 6 & -1 \\ -1 & 4 & -1 & 6 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 4B | $4$ | $90$ | $A_6:C_8$ | 2B | 4B | 4B | $ \left[ \left(\begin{array}{rrrr} 6 & 0 & -1 & -1 \\ 4 & -1 & -1 & -1 \\ -1 & -1 & 6 & 4 \\ -1 & -1 & 0 & -1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 4C | $4$ | $4050$ | $D_4:C_8$ | 2B | 4C | 4C | $ \left[ \left(\begin{array}{rrrr} 0 & 5 & 6 & 7 \\ 3 & -1 & 1 & -1 \\ -1 & -1 & 4 & 5 \\ -1 & -1 & 3 & -1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 4D | $4$ | $4050$ | $D_4:C_8$ | 2A | 4D | 4D | $ \left[ \left(\begin{array}{rrrr} 2 & 4 & 5 & 3 \\ -1 & 6 & -1 & 5 \\ 6 & 0 & 3 & 1 \\ -1 & 6 & -1 & 7 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 4E | $4$ | $8100$ | $C_4\times C_8$ | 2D | 4E | 4E | $ \left[ \left(\begin{array}{rrrr} 1 & -1 & 4 & -1 \\ 6 & 3 & 1 & 2 \\ 2 & -1 & 1 & -1 \\ 3 & 0 & 2 & 3 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 5A1 | $5$ | $72$ | $C_5\times \PGL(2,9)$ | 5A2 | 5A2 | 1A | $ \left[ \left(\begin{array}{rrrr} 1 & 2 & -1 & -1 \\ 4 & 4 & -1 & -1 \\ -1 & -1 & 1 & 6 \\ -1 & -1 & 0 & 4 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 5A2 | $5$ | $72$ | $C_5\times \PGL(2,9)$ | 5A1 | 5A1 | 1A | $ \left[ \left(\begin{array}{rrrr} -1 & 5 & -1 & -1 \\ 7 & 1 & -1 & -1 \\ -1 & -1 & -1 & 1 \\ -1 & -1 & 3 & 1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 5B1 | $5$ | $72$ | $C_5\times \PGL(2,9)$ | 5B2 | 5B2 | 1A | $ \left[ \left(\begin{array}{rrrr} 6 & -1 & 6 & -1 \\ -1 & 6 & -1 & 2 \\ 3 & -1 & 4 & -1 \\ -1 & 7 & -1 & 4 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 5B2 | $5$ | $72$ | $C_5\times \PGL(2,9)$ | 5B1 | 5B1 | 1A | $ \left[ \left(\begin{array}{rrrr} 3 & -1 & 5 & -1 \\ -1 & 3 & -1 & 1 \\ 2 & -1 & 6 & -1 \\ -1 & 6 & -1 & 6 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 5C1 | $5$ | $5184$ | $C_5\times C_{10}$ | 5C2 | 5C2 | 1A | $ \left[ \left(\begin{array}{rrrr} 7 & 5 & 6 & 0 \\ 3 & -1 & 2 & -1 \\ 1 & 7 & 6 & 0 \\ 1 & -1 & 6 & -1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 5C2 | $5$ | $5184$ | $C_5\times C_{10}$ | 5C1 | 5C1 | 1A | $ \left[ \left(\begin{array}{rrrr} 3 & 2 & 1 & 4 \\ 0 & 6 & 6 & 0 \\ 4 & 3 & 5 & 0 \\ 5 & 3 & 6 & 0 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 5D1 | $5$ | $5184$ | $C_5\times C_{10}$ | 5D2 | 5D2 | 1A | $ \left[ \left(\begin{array}{rrrr} 5 & 7 & 3 & 1 \\ 6 & 2 & 4 & 4 \\ 0 & 2 & 0 & 6 \\ 5 & 1 & 5 & 5 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 5D2 | $5$ | $5184$ | $C_5\times C_{10}$ | 5D1 | 5D1 | 1A | $ \left[ \left(\begin{array}{rrrr} 6 & 5 & 0 & 3 \\ 4 & 4 & 6 & 2 \\ 5 & 4 & 0 & 3 \\ 7 & 7 & 2 & 6 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 6A | $6$ | $3600$ | $D_4\times C_3^2$ | 3B | 2A | 6A | $ \left[ \left(\begin{array}{rrrr} -1 & -1 & 1 & 2 \\ -1 & -1 & 2 & 7 \\ 5 & 2 & -1 & -1 \\ 2 & 3 & -1 & -1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 6B | $6$ | $3600$ | $D_4\times C_3^2$ | 3A | 2B | 6B | $ \left[ \left(\begin{array}{rrrr} 2 & 7 & 7 & 0 \\ 5 & 6 & 2 & 7 \\ 3 & 0 & 5 & 6 \\ 2 & 3 & 4 & 1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 8A1 | $8$ | $3240$ | $C_8\times D_5$ | 4A | 8A3 | 8A3 | $ \left[ \left(\begin{array}{rrrr} 5 & 3 & 0 & 2 \\ 4 & 1 & 7 & 0 \\ 0 & 6 & 5 & 7 \\ 3 & 0 & 0 & 1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 8A3 | $8$ | $3240$ | $C_8\times D_5$ | 4A | 8A1 | 8A1 | $ \left[ \left(\begin{array}{rrrr} 2 & 0 & 3 & 5 \\ 1 & 6 & 2 & 3 \\ 3 & 1 & 2 & 4 \\ 6 & 3 & 5 & 6 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 8B1 | $8$ | $3240$ | $C_8\times D_5$ | 4B | 8B3 | 8B3 | $ \left[ \left(\begin{array}{rrrr} 4 & 1 & 1 & 2 \\ 5 & 5 & 2 & 6 \\ 6 & 3 & 0 & 1 \\ 3 & 3 & 5 & 1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 8B3 | $8$ | $3240$ | $C_8\times D_5$ | 4B | 8B1 | 8B1 | $ \left[ \left(\begin{array}{rrrr} 0 & 4 & 5 & 5 \\ 0 & 5 & 5 & 6 \\ 2 & 6 & 4 & 4 \\ 6 & 3 & 0 & 1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 8C1 | $8$ | $8100$ | $C_4\times C_8$ | 4E | 8C3 | 8C3 | $ \left[ \left(\begin{array}{rrrr} 6 & -1 & 2 & -1 \\ 0 & 3 & 4 & 3 \\ 0 & -1 & 6 & -1 \\ 6 & 1 & 4 & 3 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 8C3 | $8$ | $8100$ | $C_4\times C_8$ | 4E | 8C1 | 8C1 | $ \left[ \left(\begin{array}{rrrr} 6 & -1 & 0 & -1 \\ 4 & 5 & 6 & 3 \\ 6 & -1 & 6 & -1 \\ 0 & 1 & 0 & 5 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 8D1 | $8$ | $8100$ | $C_4\times C_8$ | 4E | 8D3 | 8D3 | $ \left[ \left(\begin{array}{rrrr} 5 & 6 & -1 & -1 \\ -1 & 6 & -1 & -1 \\ 3 & 4 & 2 & 7 \\ -1 & 0 & -1 & 3 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 8D3 | $8$ | $8100$ | $C_4\times C_8$ | 4E | 8D1 | 8D1 | $ \left[ \left(\begin{array}{rrrr} 3 & 2 & -1 & -1 \\ -1 & 6 & -1 & -1 \\ 5 & 4 & 2 & 5 \\ -1 & 4 & -1 & 5 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 10A1 | $10$ | $2592$ | $C_5\times D_{10}$ | 5A1 | 10A3 | 2C | $ \left[ \left(\begin{array}{rrrr} -1 & 6 & -1 & 7 \\ 1 & 0 & 6 & 1 \\ -1 & 0 & -1 & 6 \\ 7 & 6 & 1 & 4 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 10A3 | $10$ | $2592$ | $C_5\times D_{10}$ | 5A2 | 10A1 | 2C | $ \left[ \left(\begin{array}{rrrr} 4 & 4 & 1 & 5 \\ 7 & 3 & 4 & 4 \\ 6 & 6 & 0 & 4 \\ 5 & 1 & 7 & 7 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 10B1 | $10$ | $2592$ | $C_5\times D_{10}$ | 5B2 | 10B3 | 2C | $ \left[ \left(\begin{array}{rrrr} 3 & 3 & 1 & 5 \\ 5 & 7 & 3 & 1 \\ 1 & 1 & 5 & 1 \\ 7 & 1 & 3 & 1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 10B3 | $10$ | $2592$ | $C_5\times D_{10}$ | 5B1 | 10B1 | 2C | $ \left[ \left(\begin{array}{rrrr} 7 & 7 & 3 & 7 \\ 1 & 3 & 5 & 3 \\ 3 & 3 & 0 & 4 \\ 1 & 3 & 6 & 4 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 10C1 | $10$ | $3240$ | $C_5\times D_8$ | 5A1 | 10C3 | 2A | $ \left[ \left(\begin{array}{rrrr} -1 & -1 & 6 & 3 \\ -1 & -1 & 0 & 0 \\ 6 & 7 & -1 & -1 \\ 4 & 0 & -1 & -1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 10C3 | $10$ | $3240$ | $C_5\times D_8$ | 5A2 | 10C1 | 2A | $ \left[ \left(\begin{array}{rrrr} -1 & -1 & 7 & 0 \\ -1 & -1 & 5 & 4 \\ 7 & 4 & -1 & -1 \\ 1 & 4 & -1 & -1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 10D1 | $10$ | $3240$ | $C_5\times D_8$ | 5B1 | 10D3 | 2B | $ \left[ \left(\begin{array}{rrrr} 2 & 0 & 5 & 7 \\ 4 & 6 & 7 & 5 \\ 2 & 0 & 7 & 1 \\ 0 & 2 & 5 & 3 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 10D3 | $10$ | $3240$ | $C_5\times D_8$ | 5B2 | 10D1 | 2B | $ \left[ \left(\begin{array}{rrrr} 0 & 6 & 6 & 0 \\ 2 & 4 & 0 & 6 \\ 3 & 1 & 2 & 4 \\ 1 & 3 & 0 & 6 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 10E1 | $10$ | $5184$ | $C_5\times C_{10}$ | 5C1 | 10E3 | 2C | $ \left[ \left(\begin{array}{rrrr} 7 & 4 & 1 & 2 \\ 2 & 5 & 4 & 3 \\ 4 & 1 & 2 & 3 \\ 3 & 6 & 1 & 0 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 10E3 | $10$ | $5184$ | $C_5\times C_{10}$ | 5C2 | 10E1 | 2C | $ \left[ \left(\begin{array}{rrrr} -1 & -1 & 6 & 6 \\ -1 & -1 & 0 & 1 \\ 1 & 5 & 1 & 1 \\ 7 & 4 & 7 & 0 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 10F1 | $10$ | $5184$ | $C_5\times C_{10}$ | 5D1 | 10F3 | 2C | $ \left[ \left(\begin{array}{rrrr} 4 & 7 & 7 & 6 \\ 6 & 5 & 1 & 4 \\ 4 & 7 & 3 & 2 \\ 2 & 1 & 1 & 4 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 10F3 | $10$ | $5184$ | $C_5\times C_{10}$ | 5D2 | 10F1 | 2C | $ \left[ \left(\begin{array}{rrrr} -1 & 5 & -1 & 6 \\ 4 & 1 & 1 & 2 \\ -1 & 7 & -1 & -1 \\ 2 & 7 & -1 & -1 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 12A | $12$ | $7200$ | $C_3\times C_{12}$ | 6A | 4A | 12A | $ \left[ \left(\begin{array}{rrrr} 5 & 4 & 1 & 4 \\ 0 & 3 & 4 & 3 \\ 5 & 4 & 5 & 0 \\ 4 & 7 & 4 & 3 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 12B | $12$ | $7200$ | $C_3\times C_{12}$ | 6B | 4B | 12B | $ \left[ \left(\begin{array}{rrrr} -1 & 0 & -1 & 2 \\ 6 & 3 & 4 & 5 \\ -1 & 2 & -1 & 1 \\ 4 & 1 & 7 & 0 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 15A1 | $15$ | $5760$ | $C_3\times C_{15}$ | 15A2 | 5A1 | 3A | $ \left[ \left(\begin{array}{rrrr} -1 & 3 & -1 & 0 \\ 5 & 7 & 6 & 4 \\ -1 & 0 & -1 & 2 \\ 6 & 0 & 4 & 2 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 15A2 | $15$ | $5760$ | $C_3\times C_{15}$ | 15A1 | 5A2 | 3A | $ \left[ \left(\begin{array}{rrrr} 5 & 7 & 7 & 5 \\ 1 & 2 & 3 & 0 \\ 3 & 5 & 2 & 0 \\ 3 & 4 & 2 & 7 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 15B1 | $15$ | $5760$ | $C_3\times C_{15}$ | 15B2 | 5B2 | 3B | $ \left[ \left(\begin{array}{rrrr} 6 & 0 & 2 & 0 \\ -1 & 6 & -1 & 6 \\ 3 & 5 & 5 & 3 \\ -1 & 7 & -1 & 5 \end{array}\right) \right] $ |
| $\PSOPlus(4,9)$ | 15B2 | $15$ | $5760$ | $C_3\times C_{15}$ | 15B1 | 5B1 | 3B | $ \left[ \left(\begin{array}{rrrr} 6 & 4 & 1 & 3 \\ -1 & 6 & -1 & 5 \\ 2 & 0 & 0 & 2 \\ -1 & 6 & -1 & 0 \end{array}\right) \right] $ |