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Elements of the group are displayed as matrices in $\GL_{2}(\Z/{30}\Z)$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 3P | ||||||
| $S_3\times D_6^2$ | 1A | $1$ | $1$ | $S_3\times D_6^2$ | 1A | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2A | $2$ | $1$ | $S_3\times D_6^2$ | 1A | 2A | $ \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2B | $2$ | $1$ | $S_3\times D_6^2$ | 1A | 2B | $ \left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2C | $2$ | $1$ | $S_3\times D_6^2$ | 1A | 2C | $ \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2D | $2$ | $3$ | $C_2\times D_6^2$ | 1A | 2D | $ \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2E | $2$ | $3$ | $C_2\times D_6^2$ | 1A | 2E | $ \left(\begin{array}{rr} 26 & 15 \\ 15 & 26 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2F | $2$ | $3$ | $C_2\times D_6^2$ | 1A | 2F | $ \left(\begin{array}{rr} 19 & 12 \\ 0 & 1 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2G | $2$ | $3$ | $C_2\times D_6^2$ | 1A | 2G | $ \left(\begin{array}{rr} 29 & 12 \\ 0 & 11 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2H | $2$ | $3$ | $C_2\times D_6^2$ | 1A | 2H | $ \left(\begin{array}{rr} 11 & 20 \\ 0 & 1 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2I | $2$ | $3$ | $C_2\times D_6^2$ | 1A | 2I | $ \left(\begin{array}{rr} 1 & 10 \\ 0 & 11 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2J | $2$ | $3$ | $C_2\times D_6^2$ | 1A | 2J | $ \left(\begin{array}{rr} 19 & 10 \\ 0 & 29 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2K | $2$ | $3$ | $C_2\times D_6^2$ | 1A | 2K | $ \left(\begin{array}{rr} 29 & 15 \\ 0 & 29 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2L | $2$ | $3$ | $C_2\times D_6^2$ | 1A | 2L | $ \left(\begin{array}{rr} 11 & 0 \\ 24 & 29 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2M | $2$ | $3$ | $C_2\times D_6^2$ | 1A | 2M | $ \left(\begin{array}{rr} 1 & 0 \\ 24 & 19 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2N | $2$ | $3$ | $C_2\times D_6^2$ | 1A | 2N | $ \left(\begin{array}{rr} 4 & 15 \\ 15 & 4 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2O | $2$ | $3$ | $C_2\times D_6^2$ | 1A | 2O | $ \left(\begin{array}{rr} 29 & 20 \\ 0 & 19 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2P | $2$ | $9$ | $C_2^3\times D_6$ | 1A | 2P | $ \left(\begin{array}{rr} 19 & 27 \\ 0 & 1 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2Q | $2$ | $9$ | $C_2^3\times D_6$ | 1A | 2Q | $ \left(\begin{array}{rr} 14 & 27 \\ 15 & 26 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2R | $2$ | $9$ | $C_2^3\times D_6$ | 1A | 2R | $ \left(\begin{array}{rr} 11 & 15 \\ 24 & 29 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2S | $2$ | $9$ | $C_2^3\times D_6$ | 1A | 2S | $ \left(\begin{array}{rr} 11 & 25 \\ 0 & 1 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2T | $2$ | $9$ | $C_2^3\times D_6$ | 1A | 2T | $ \left(\begin{array}{rr} 19 & 5 \\ 0 & 29 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2U | $2$ | $9$ | $C_2^3\times D_6$ | 1A | 2U | $ \left(\begin{array}{rr} 29 & 2 \\ 0 & 1 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2V | $2$ | $9$ | $C_2^3\times D_6$ | 1A | 2V | $ \left(\begin{array}{rr} 19 & 22 \\ 0 & 11 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2W | $2$ | $9$ | $C_2^3\times D_6$ | 1A | 2W | $ \left(\begin{array}{rr} 16 & 15 \\ 9 & 4 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2X | $2$ | $9$ | $C_2^3\times D_6$ | 1A | 2X | $ \left(\begin{array}{rr} 16 & 5 \\ 15 & 26 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2Y | $2$ | $9$ | $C_2^3\times D_6$ | 1A | 2Y | $ \left(\begin{array}{rr} 14 & 25 \\ 15 & 4 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2Z | $2$ | $9$ | $C_2^3\times D_6$ | 1A | 2Z | $ \left(\begin{array}{rr} 1 & 10 \\ 24 & 29 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2AA | $2$ | $9$ | $C_2^3\times D_6$ | 1A | 2AA | $ \left(\begin{array}{rr} 11 & 20 \\ 24 & 19 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2AB | $2$ | $27$ | $C_2^5$ | 1A | 2AB | $ \left(\begin{array}{rr} 29 & 7 \\ 0 & 1 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2AC | $2$ | $27$ | $C_2^5$ | 1A | 2AC | $ \left(\begin{array}{rr} 4 & 17 \\ 15 & 26 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2AD | $2$ | $27$ | $C_2^5$ | 1A | 2AD | $ \left(\begin{array}{rr} 1 & 5 \\ 24 & 29 \end{array}\right) $ |
| $S_3\times D_6^2$ | 2AE | $2$ | $27$ | $C_2^5$ | 1A | 2AE | $ \left(\begin{array}{rr} 26 & 25 \\ 9 & 4 \end{array}\right) $ |
| $S_3\times D_6^2$ | 3A | $3$ | $2$ | $C_3\times D_6^2$ | 3A | 1A | $ \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right) $ |
| $S_3\times D_6^2$ | 3B | $3$ | $2$ | $C_3\times D_6^2$ | 3B | 1A | $ \left(\begin{array}{rr} 1 & 18 \\ 24 & 13 \end{array}\right) $ |
| $S_3\times D_6^2$ | 3C | $3$ | $2$ | $C_3\times D_6^2$ | 3C | 1A | $ \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right) $ |
| $S_3\times D_6^2$ | 3D | $3$ | $4$ | $S_3\times C_6^2$ | 3D | 1A | $ \left(\begin{array}{rr} 16 & 25 \\ 15 & 1 \end{array}\right) $ |
| $S_3\times D_6^2$ | 3E | $3$ | $4$ | $S_3\times C_6^2$ | 3E | 1A | $ \left(\begin{array}{rr} 28 & 27 \\ 21 & 1 \end{array}\right) $ |
| $S_3\times D_6^2$ | 3F | $3$ | $4$ | $S_3\times C_6^2$ | 3F | 1A | $ \left(\begin{array}{rr} 1 & 28 \\ 24 & 13 \end{array}\right) $ |
| $S_3\times D_6^2$ | 3G | $3$ | $8$ | $C_3\times C_6^2$ | 3G | 1A | $ \left(\begin{array}{rr} 28 & 7 \\ 21 & 1 \end{array}\right) $ |
| $S_3\times D_6^2$ | 6A | $6$ | $2$ | $C_3\times D_6^2$ | 3A | 2A | $ \left(\begin{array}{rr} 11 & 15 \\ 15 & 26 \end{array}\right) $ |
| $S_3\times D_6^2$ | 6B | $6$ | $2$ | $C_3\times D_6^2$ | 3B | 2B | $ \left(\begin{array}{rr} 17 & 18 \\ 24 & 29 \end{array}\right) $ |
| $S_3\times D_6^2$ | 6C | $6$ | $2$ | $C_3\times D_6^2$ | 3C | 2A | $ \left(\begin{array}{rr} 11 & 10 \\ 0 & 11 \end{array}\right) $ |
| $S_3\times D_6^2$ | 6D | $6$ | $2$ | $C_3\times D_6^2$ | 3B | 2A | $ \left(\begin{array}{rr} 11 & 18 \\ 24 & 23 \end{array}\right) $ |
| $S_3\times D_6^2$ | 6E | $6$ | $2$ | $C_3\times D_6^2$ | 3B | 2C | $ \left(\begin{array}{rr} 7 & 18 \\ 24 & 19 \end{array}\right) $ |
| $S_3\times D_6^2$ | 6F | $6$ | $2$ | $C_3\times D_6^2$ | 3A | 2B | $ \left(\begin{array}{rr} 14 & 15 \\ 15 & 29 \end{array}\right) $ |
| $S_3\times D_6^2$ | 6G | $6$ | $2$ | $C_3\times D_6^2$ | 3A | 2C | $ \left(\begin{array}{rr} 19 & 15 \\ 15 & 4 \end{array}\right) $ |
| $S_3\times D_6^2$ | 6H | $6$ | $2$ | $C_3\times D_6^2$ | 3C | 2B | $ \left(\begin{array}{rr} 29 & 10 \\ 0 & 29 \end{array}\right) $ |
| $S_3\times D_6^2$ | 6I | $6$ | $2$ | $C_3\times D_6^2$ | 3C | 2C | $ \left(\begin{array}{rr} 19 & 20 \\ 0 & 19 \end{array}\right) $ |
| $S_3\times D_6^2$ | 6J | $6$ | $4$ | $S_3\times C_6^2$ | 3D | 2A | $ \left(\begin{array}{rr} 11 & 25 \\ 15 & 26 \end{array}\right) $ |
| $S_3\times D_6^2$ | 6K | $6$ | $4$ | $S_3\times C_6^2$ | 3E | 2A | $ \left(\begin{array}{rr} 11 & 3 \\ 9 & 8 \end{array}\right) $ |