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Elements of the group are displayed as matrices in $\GL_{2}(\Z/{28}\Z)$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 3P | ||||||
| $C_6\times D_6:S_4$ | 1A | $1$ | $1$ | $C_6\times D_6:S_4$ | 1A | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 2A | $2$ | $1$ | $C_6\times D_6:S_4$ | 1A | 2A | $ \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 2B | $2$ | $1$ | $C_6\times D_6:S_4$ | 1A | 2B | $ \left(\begin{array}{rr} 27 & 0 \\ 0 & 27 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 2C | $2$ | $1$ | $C_6\times D_6:S_4$ | 1A | 2C | $ \left(\begin{array}{rr} 15 & 0 \\ 0 & 15 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 2D | $2$ | $3$ | $C_6^2.C_2^4$ | 1A | 2D | $ \left(\begin{array}{rr} 27 & 14 \\ 0 & 27 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 2E | $2$ | $3$ | $C_6^2.C_2^4$ | 1A | 2E | $ \left(\begin{array}{rr} 13 & 14 \\ 0 & 13 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 2F | $2$ | $3$ | $C_6^2.C_2^4$ | 1A | 2F | $ \left(\begin{array}{rr} 15 & 0 \\ 14 & 15 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 2G | $2$ | $3$ | $C_6^2.C_2^4$ | 1A | 2G | $ \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 2H | $2$ | $6$ | $C_2^3:C_6^2$ | 1A | 2H | $ \left(\begin{array}{rr} 7 & 2 \\ 18 & 21 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 2I | $2$ | $6$ | $C_2^3:C_6^2$ | 1A | 2I | $ \left(\begin{array}{rr} 7 & 26 \\ 10 & 21 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 2J | $2$ | $18$ | $C_2^4\times C_6$ | 1A | 2J | $ \left(\begin{array}{rr} 21 & 16 \\ 18 & 7 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 2K | $2$ | $18$ | $C_2^4\times C_6$ | 1A | 2K | $ \left(\begin{array}{rr} 21 & 12 \\ 10 & 7 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 2L | $2$ | $36$ | $C_2^3\times C_6$ | 1A | 2L | $ \left(\begin{array}{rr} 7 & 1 \\ 8 & 21 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 2M | $2$ | $36$ | $C_2^3\times C_6$ | 1A | 2M | $ \left(\begin{array}{rr} 7 & 20 \\ 13 & 21 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 3A1 | $3$ | $1$ | $C_6\times D_6:S_4$ | 3A-1 | 1A | $ \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 3A-1 | $3$ | $1$ | $C_6\times D_6:S_4$ | 3A1 | 1A | $ \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 3B | $3$ | $2$ | $C_6^2.S_4$ | 3B | 1A | $ \left(\begin{array}{rr} 25 & 0 \\ 0 & 9 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 3C1 | $3$ | $2$ | $C_6^2.S_4$ | 3C-1 | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 25 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 3C-1 | $3$ | $2$ | $C_6^2.S_4$ | 3C1 | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 9 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 3D | $3$ | $8$ | $S_3\times C_6^2$ | 3D | 1A | $ \left(\begin{array}{rr} 1 & 7 \\ 7 & 22 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 3E1 | $3$ | $8$ | $S_3\times C_6^2$ | 3E-1 | 1A | $ \left(\begin{array}{rr} 9 & 7 \\ 7 & 2 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 3E-1 | $3$ | $8$ | $S_3\times C_6^2$ | 3E1 | 1A | $ \left(\begin{array}{rr} 18 & 21 \\ 21 & 25 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 3F | $3$ | $16$ | $C_3\times C_6^2$ | 3F | 1A | $ \left(\begin{array}{rr} 25 & 7 \\ 7 & 2 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 3G1 | $3$ | $16$ | $C_3\times C_6^2$ | 3G-1 | 1A | $ \left(\begin{array}{rr} 1 & 7 \\ 7 & 18 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 3G-1 | $3$ | $16$ | $C_3\times C_6^2$ | 3G1 | 1A | $ \left(\begin{array}{rr} 22 & 21 \\ 21 & 9 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 4A | $4$ | $12$ | $C_2\times C_6\times C_{12}$ | 2C | 4A | $ \left(\begin{array}{rr} 15 & 7 \\ 14 & 1 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 4B | $4$ | $12$ | $C_2\times C_6\times C_{12}$ | 2G | 4B | $ \left(\begin{array}{rr} 1 & 21 \\ 0 & 1 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 4C | $4$ | $12$ | $C_2\times C_6\times C_{12}$ | 2C | 4C | $ \left(\begin{array}{rr} 13 & 14 \\ 21 & 27 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 4D | $4$ | $12$ | $C_2\times C_6\times C_{12}$ | 2G | 4D | $ \left(\begin{array}{rr} 13 & 0 \\ 7 & 13 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 4E | $4$ | $36$ | $C_2^2\times C_{12}$ | 2F | 4E | $ \left(\begin{array}{rr} 21 & 15 \\ 22 & 21 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 4F | $4$ | $36$ | $C_2^2\times C_{12}$ | 2F | 4F | $ \left(\begin{array}{rr} 7 & 6 \\ 27 & 7 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6A1 | $6$ | $1$ | $C_6\times D_6:S_4$ | 3A1 | 2A | $ \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6A-1 | $6$ | $1$ | $C_6\times D_6:S_4$ | 3A-1 | 2A | $ \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6B1 | $6$ | $1$ | $C_6\times D_6:S_4$ | 3A1 | 2B | $ \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6B-1 | $6$ | $1$ | $C_6\times D_6:S_4$ | 3A-1 | 2B | $ \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6C1 | $6$ | $1$ | $C_6\times D_6:S_4$ | 3A-1 | 2C | $ \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6C-1 | $6$ | $1$ | $C_6\times D_6:S_4$ | 3A1 | 2C | $ \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6D | $6$ | $2$ | $C_6^2.S_4$ | 3B | 2A | $ \left(\begin{array}{rr} 5 & 0 \\ 0 & 17 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6E | $6$ | $2$ | $C_6^2.S_4$ | 3B | 2B | $ \left(\begin{array}{rr} 3 & 0 \\ 0 & 19 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6F | $6$ | $2$ | $C_6^2.S_4$ | 3B | 2C | $ \left(\begin{array}{rr} 11 & 0 \\ 0 & 23 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6G1 | $6$ | $2$ | $C_6^2.S_4$ | 3C1 | 2A | $ \left(\begin{array}{rr} 13 & 0 \\ 0 & 5 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6G-1 | $6$ | $2$ | $C_6^2.S_4$ | 3C-1 | 2A | $ \left(\begin{array}{rr} 17 & 0 \\ 0 & 13 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6H1 | $6$ | $2$ | $C_6^2.S_4$ | 3C1 | 2B | $ \left(\begin{array}{rr} 19 & 0 \\ 0 & 27 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6H-1 | $6$ | $2$ | $C_6^2.S_4$ | 3C-1 | 2B | $ \left(\begin{array}{rr} 27 & 0 \\ 0 & 3 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6I1 | $6$ | $2$ | $C_6^2.S_4$ | 3C-1 | 2C | $ \left(\begin{array}{rr} 15 & 0 \\ 0 & 11 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6I-1 | $6$ | $2$ | $C_6^2.S_4$ | 3C1 | 2C | $ \left(\begin{array}{rr} 15 & 0 \\ 0 & 23 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6J1 | $6$ | $3$ | $C_6^2.C_2^4$ | 3A-1 | 2D | $ \left(\begin{array}{rr} 19 & 14 \\ 0 & 19 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6J-1 | $6$ | $3$ | $C_6^2.C_2^4$ | 3A1 | 2D | $ \left(\begin{array}{rr} 3 & 0 \\ 14 & 3 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6K1 | $6$ | $3$ | $C_6^2.C_2^4$ | 3A-1 | 2E | $ \left(\begin{array}{rr} 5 & 14 \\ 0 & 5 \end{array}\right) $ |
| $C_6\times D_6:S_4$ | 6K-1 | $6$ | $3$ | $C_6^2.C_2^4$ | 3A1 | 2E | $ \left(\begin{array}{rr} 17 & 0 \\ 14 & 17 \end{array}\right) $ |