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Elements of the group are displayed as matrices in $\GL_{4}(\F_{3})$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 3P | ||||||
| $C_3^4:C_2^3$ | 1A | $1$ | $1$ | $C_3^4:C_2^3$ | 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) $ |
| $C_3^4:C_2^3$ | 2A | $2$ | $1$ | $C_3^4:C_2^3$ | 1A | 2A | $ \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 2B | $2$ | $9$ | $C_6:D_6$ | 1A | 2B | $ \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 2 & 2 & 1 \\ 0 & 1 & 0 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 2C | $2$ | $9$ | $C_6:D_6$ | 1A | 2C | $ \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 2 \\ 0 & 2 & 0 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 2D | $2$ | $9$ | $C_6:D_6$ | 1A | 2D | $ \left(\begin{array}{rrrr} 1 & 1 & 0 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 2 & 0 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 2E | $2$ | $9$ | $C_6:D_6$ | 1A | 2E | $ \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 2F | $2$ | $81$ | $C_2^3$ | 1A | 2F | $ \left(\begin{array}{rrrr} 1 & 0 & 1 & 0 \\ 2 & 0 & 0 & 2 \\ 0 & 0 & 2 & 0 \\ 2 & 2 & 2 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 2G | $2$ | $81$ | $C_2^3$ | 1A | 2G | $ \left(\begin{array}{rrrr} 1 & 2 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 2 & 2 & 0 & 1 \\ 2 & 2 & 2 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3A | $3$ | $2$ | $C_3^2:C_6^2$ | 3A | 1A | $ \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 2 & 2 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 2 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3B | $3$ | $2$ | $C_3^2:C_6^2$ | 3B | 1A | $ \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 2 & 0 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3C | $3$ | $2$ | $C_3^2:C_6^2$ | 3C | 1A | $ \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 2 & 1 & 1 \\ 2 & 1 & 0 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3D | $3$ | $2$ | $C_3^2:C_6^2$ | 3D | 1A | $ \left(\begin{array}{rrrr} 1 & 1 & 2 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 2 \\ 0 & 2 & 1 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3E | $3$ | $2$ | $C_3^2:C_6^2$ | 3E | 1A | $ \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 2 & 2 & 2 \\ 1 & 2 & 1 & 1 \\ 2 & 2 & 2 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3F | $3$ | $2$ | $C_3^2:C_6^2$ | 3F | 1A | $ \left(\begin{array}{rrrr} 1 & 2 & 1 & 1 \\ 2 & 2 & 0 & 2 \\ 0 & 2 & 2 & 1 \\ 2 & 2 & 2 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3G | $3$ | $2$ | $C_3^2:C_6^2$ | 3G | 1A | $ \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 2 & 1 & 1 \\ 2 & 0 & 1 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3H | $3$ | $2$ | $C_3^2:C_6^2$ | 3H | 1A | $ \left(\begin{array}{rrrr} 1 & 1 & 2 & 2 \\ 2 & 2 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 2 & 0 & 1 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3I | $3$ | $4$ | $C_3^3\times C_6$ | 3I | 1A | $ \left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 2 & 0 & 1 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3J | $3$ | $4$ | $C_3^3\times C_6$ | 3J | 1A | $ \left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 1 & 1 & 2 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3K | $3$ | $4$ | $C_3^3\times C_6$ | 3K | 1A | $ \left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 0 & 2 & 2 & 2 \\ 1 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3L | $3$ | $4$ | $C_3^3\times C_6$ | 3L | 1A | $ \left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 2 & 1 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3M | $3$ | $4$ | $C_3^3\times C_6$ | 3M | 1A | $ \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 1 & 2 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 2 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3N | $3$ | $4$ | $C_3^3\times C_6$ | 3N | 1A | $ \left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ 2 & 2 & 2 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3O | $3$ | $4$ | $C_3^3\times C_6$ | 3O | 1A | $ \left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 1 & 2 & 1 & 2 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 2 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3P | $3$ | $4$ | $C_3^3\times C_6$ | 3P | 1A | $ \left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 2 & 2 & 0 & 2 \\ 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3Q | $3$ | $4$ | $C_3^3\times C_6$ | 3Q | 1A | $ \left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 2 & 0 & 2 & 1 \\ 1 & 0 & 0 & 0 \\ 1 & 2 & 0 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3R | $3$ | $4$ | $C_3^3\times C_6$ | 3R | 1A | $ \left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 2 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 2 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3S | $3$ | $4$ | $C_3^3\times C_6$ | 3S | 1A | $ \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 2 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3T | $3$ | $4$ | $C_3^3\times C_6$ | 3T | 1A | $ \left(\begin{array}{rrrr} 1 & 1 & 2 & 2 \\ 0 & 2 & 2 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3U | $3$ | $4$ | $C_3^3\times C_6$ | 3U | 1A | $ \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 1 & 2 & 1 & 2 \\ 1 & 2 & 1 & 1 \\ 0 & 2 & 1 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3V | $3$ | $4$ | $C_3^3\times C_6$ | 3V | 1A | $ \left(\begin{array}{rrrr} 1 & 2 & 1 & 1 \\ 1 & 2 & 1 & 2 \\ 0 & 2 & 2 & 1 \\ 1 & 2 & 0 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3W | $3$ | $4$ | $C_3^3\times C_6$ | 3W | 1A | $ \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 1 & 1 & 2 & 0 \\ 1 & 2 & 1 & 1 \\ 0 & 1 & 2 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 3X | $3$ | $4$ | $C_3^3\times C_6$ | 3X | 1A | $ \left(\begin{array}{rrrr} 1 & 1 & 2 & 2 \\ 1 & 1 & 2 & 0 \\ 0 & 1 & 0 & 2 \\ 1 & 2 & 0 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6A | $6$ | $2$ | $C_3^2:C_6^2$ | 3A | 2A | $ \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 2 \\ 0 & 0 & 2 & 0 \\ 0 & 1 & 2 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6B | $6$ | $2$ | $C_3^2:C_6^2$ | 3B | 2A | $ \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 2 & 0 \\ 1 & 2 & 0 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6C | $6$ | $2$ | $C_3^2:C_6^2$ | 3C | 2A | $ \left(\begin{array}{rrrr} 0 & 2 & 0 & 1 \\ 0 & 2 & 0 & 0 \\ 1 & 2 & 2 & 1 \\ 2 & 1 & 0 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6D | $6$ | $2$ | $C_3^2:C_6^2$ | 3D | 2A | $ \left(\begin{array}{rrrr} 2 & 1 & 2 & 2 \\ 0 & 2 & 0 & 0 \\ 0 & 1 & 1 & 2 \\ 0 & 2 & 1 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6E | $6$ | $2$ | $C_3^2:C_6^2$ | 3E | 2A | $ \left(\begin{array}{rrrr} 0 & 2 & 0 & 1 \\ 0 & 0 & 2 & 2 \\ 1 & 2 & 2 & 1 \\ 2 & 2 & 2 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6F | $6$ | $2$ | $C_3^2:C_6^2$ | 3F | 2A | $ \left(\begin{array}{rrrr} 2 & 2 & 1 & 1 \\ 2 & 0 & 0 & 2 \\ 0 & 2 & 0 & 1 \\ 2 & 2 & 2 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6G | $6$ | $2$ | $C_3^2:C_6^2$ | 3G | 2A | $ \left(\begin{array}{rrrr} 0 & 2 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 1 & 2 & 2 & 1 \\ 2 & 0 & 1 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6H | $6$ | $2$ | $C_3^2:C_6^2$ | 3H | 2A | $ \left(\begin{array}{rrrr} 2 & 1 & 2 & 2 \\ 2 & 0 & 0 & 2 \\ 0 & 1 & 1 & 2 \\ 2 & 0 & 1 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6I | $6$ | $4$ | $C_3^3\times C_6$ | 3I | 2A | $ \left(\begin{array}{rrrr} 0 & 0 & 2 & 0 \\ 0 & 2 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 2 & 0 & 1 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6J | $6$ | $4$ | $C_3^3\times C_6$ | 3J | 2A | $ \left(\begin{array}{rrrr} 0 & 0 & 2 & 0 \\ 1 & 2 & 2 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6K | $6$ | $4$ | $C_3^3\times C_6$ | 3K | 2A | $ \left(\begin{array}{rrrr} 0 & 0 & 2 & 0 \\ 0 & 0 & 2 & 2 \\ 1 & 0 & 1 & 0 \\ 2 & 1 & 0 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6L | $6$ | $4$ | $C_3^3\times C_6$ | 3L | 2A | $ \left(\begin{array}{rrrr} 0 & 0 & 2 & 0 \\ 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 2 & 1 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6M | $6$ | $4$ | $C_3^3\times C_6$ | 3M | 2A | $ \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 1 & 2 & 2 & 0 \\ 0 & 0 & 2 & 0 \\ 1 & 0 & 2 & 2 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6N | $6$ | $4$ | $C_3^3\times C_6$ | 3N | 2A | $ \left(\begin{array}{rrrr} 0 & 0 & 2 & 0 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 2 & 2 & 2 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6O | $6$ | $4$ | $C_3^3\times C_6$ | 3O | 2A | $ \left(\begin{array}{rrrr} 0 & 0 & 2 & 0 \\ 1 & 0 & 1 & 2 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 2 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6P | $6$ | $4$ | $C_3^3\times C_6$ | 3P | 2A | $ \left(\begin{array}{rrrr} 0 & 0 & 2 & 0 \\ 2 & 0 & 0 & 2 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6Q | $6$ | $4$ | $C_3^3\times C_6$ | 3Q | 2A | $ \left(\begin{array}{rrrr} 0 & 0 & 2 & 0 \\ 2 & 1 & 2 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 2 & 0 & 0 \end{array}\right) $ |
| $C_3^4:C_2^3$ | 6R | $6$ | $4$ | $C_3^3\times C_6$ | 3R | 2A | $ \left(\begin{array}{rrrr} 0 & 0 & 2 & 0 \\ 2 & 2 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 2 & 2 \end{array}\right) $ |