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Download displayed columns for resultsElements of the group are displayed as matrices in $\GL_{2}(\F_{31})$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | ||
|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 5P | ||||||
| $C_{10}.S_4$ | 1A | $1$ | $1$ | $C_{10}.S_4$ | 1A | 1A | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right) $ |
| $C_{10}.S_4$ | 2A | $2$ | $1$ | $C_{10}.S_4$ | 1A | 2A | 2A | $ \left(\begin{array}{rr} 30 & 0 \\ 0 & 30 \end{array}\right) $ |
| $C_{10}.S_4$ | 3A | $3$ | $8$ | $C_{30}$ | 3A | 1A | 3A | $ \left(\begin{array}{rr} 0 & 16 \\ 29 & 30 \end{array}\right) $ |
| $C_{10}.S_4$ | 4A | $4$ | $6$ | $C_{40}$ | 2A | 4A | 4A | $ \left(\begin{array}{rr} 14 & 19 \\ 19 & 17 \end{array}\right) $ |
| $C_{10}.S_4$ | 4B | $4$ | $12$ | $C_{20}$ | 2A | 4B | 4B | $ \left(\begin{array}{rr} 4 & 20 \\ 10 & 27 \end{array}\right) $ |
| $C_{10}.S_4$ | 5A1 | $5$ | $1$ | $C_{10}.S_4$ | 5A2 | 5A-2 | 1A | $ \left(\begin{array}{rr} 4 & 0 \\ 0 & 4 \end{array}\right) $ |
| $C_{10}.S_4$ | 5A-1 | $5$ | $1$ | $C_{10}.S_4$ | 5A-2 | 5A2 | 1A | $ \left(\begin{array}{rr} 8 & 0 \\ 0 & 8 \end{array}\right) $ |
| $C_{10}.S_4$ | 5A2 | $5$ | $1$ | $C_{10}.S_4$ | 5A-1 | 5A1 | 1A | $ \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right) $ |
| $C_{10}.S_4$ | 5A-2 | $5$ | $1$ | $C_{10}.S_4$ | 5A1 | 5A-1 | 1A | $ \left(\begin{array}{rr} 2 & 0 \\ 0 & 2 \end{array}\right) $ |
| $C_{10}.S_4$ | 6A | $6$ | $8$ | $C_{30}$ | 3A | 2A | 6A | $ \left(\begin{array}{rr} 1 & 16 \\ 29 & 0 \end{array}\right) $ |
| $C_{10}.S_4$ | 8A1 | $8$ | $6$ | $C_{40}$ | 4A | 8A3 | 8A3 | $ \left(\begin{array}{rr} 29 & 14 \\ 14 & 10 \end{array}\right) $ |
| $C_{10}.S_4$ | 8A3 | $8$ | $6$ | $C_{40}$ | 4A | 8A1 | 8A1 | $ \left(\begin{array}{rr} 21 & 14 \\ 14 & 2 \end{array}\right) $ |
| $C_{10}.S_4$ | 10A1 | $10$ | $1$ | $C_{10}.S_4$ | 5A1 | 10A3 | 2A | $ \left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right) $ |
| $C_{10}.S_4$ | 10A-1 | $10$ | $1$ | $C_{10}.S_4$ | 5A-1 | 10A-3 | 2A | $ \left(\begin{array}{rr} 15 & 0 \\ 0 & 15 \end{array}\right) $ |
| $C_{10}.S_4$ | 10A3 | $10$ | $1$ | $C_{10}.S_4$ | 5A-2 | 10A-1 | 2A | $ \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right) $ |
| $C_{10}.S_4$ | 10A-3 | $10$ | $1$ | $C_{10}.S_4$ | 5A2 | 10A1 | 2A | $ \left(\begin{array}{rr} 27 & 0 \\ 0 & 27 \end{array}\right) $ |
| $C_{10}.S_4$ | 15A1 | $15$ | $8$ | $C_{30}$ | 15A2 | 5A2 | 3A | $ \left(\begin{array}{rr} 23 & 27 \\ 16 & 0 \end{array}\right) $ |
| $C_{10}.S_4$ | 15A-1 | $15$ | $8$ | $C_{30}$ | 15A-2 | 5A-2 | 3A | $ \left(\begin{array}{rr} 0 & 2 \\ 23 & 27 \end{array}\right) $ |
| $C_{10}.S_4$ | 15A2 | $15$ | $8$ | $C_{30}$ | 15A-1 | 5A-1 | 3A | $ \left(\begin{array}{rr} 0 & 1 \\ 27 & 29 \end{array}\right) $ |
| $C_{10}.S_4$ | 15A-2 | $15$ | $8$ | $C_{30}$ | 15A1 | 5A1 | 3A | $ \left(\begin{array}{rr} 15 & 23 \\ 1 & 0 \end{array}\right) $ |
| $C_{10}.S_4$ | 20A1 | $20$ | $6$ | $C_{40}$ | 10A1 | 20A3 | 4A | $ \left(\begin{array}{rr} 19 & 28 \\ 28 & 12 \end{array}\right) $ |
| $C_{10}.S_4$ | 20A-1 | $20$ | $6$ | $C_{40}$ | 10A-1 | 20A-3 | 4A | $ \left(\begin{array}{rr} 6 & 17 \\ 17 & 25 \end{array}\right) $ |
| $C_{10}.S_4$ | 20A3 | $20$ | $6$ | $C_{40}$ | 10A3 | 20A-1 | 4A | $ \left(\begin{array}{rr} 24 & 6 \\ 6 & 7 \end{array}\right) $ |
| $C_{10}.S_4$ | 20A-3 | $20$ | $6$ | $C_{40}$ | 10A-3 | 20A1 | 4A | $ \left(\begin{array}{rr} 28 & 7 \\ 7 & 3 \end{array}\right) $ |
| $C_{10}.S_4$ | 20B1 | $20$ | $12$ | $C_{20}$ | 10A1 | 20B3 | 4B | $ \left(\begin{array}{rr} 1 & 5 \\ 18 & 30 \end{array}\right) $ |
| $C_{10}.S_4$ | 20B-1 | $20$ | $12$ | $C_{20}$ | 10A-1 | 20B-3 | 4B | $ \left(\begin{array}{rr} 3 & 11 \\ 9 & 28 \end{array}\right) $ |
| $C_{10}.S_4$ | 20B3 | $20$ | $12$ | $C_{20}$ | 10A3 | 20B-1 | 4B | $ \left(\begin{array}{rr} 1 & 22 \\ 1 & 30 \end{array}\right) $ |
| $C_{10}.S_4$ | 20B-3 | $20$ | $12$ | $C_{20}$ | 10A-3 | 20B1 | 4B | $ \left(\begin{array}{rr} 4 & 26 \\ 4 & 27 \end{array}\right) $ |
| $C_{10}.S_4$ | 30A1 | $30$ | $8$ | $C_{30}$ | 15A1 | 10A-3 | 6A | $ \left(\begin{array}{rr} 0 & 23 \\ 1 & 16 \end{array}\right) $ |
| $C_{10}.S_4$ | 30A-1 | $30$ | $8$ | $C_{30}$ | 15A-1 | 10A3 | 6A | $ \left(\begin{array}{rr} 0 & 30 \\ 4 & 2 \end{array}\right) $ |
| $C_{10}.S_4$ | 30A7 | $30$ | $8$ | $C_{30}$ | 15A2 | 10A-1 | 6A | $ \left(\begin{array}{rr} 0 & 27 \\ 16 & 8 \end{array}\right) $ |
| $C_{10}.S_4$ | 30A-7 | $30$ | $8$ | $C_{30}$ | 15A-2 | 10A1 | 6A | $ \left(\begin{array}{rr} 0 & 29 \\ 8 & 4 \end{array}\right) $ |
| $C_{10}.S_4$ | 40A1 | $40$ | $6$ | $C_{40}$ | 20A1 | 40A3 | 8A1 | $ \left(\begin{array}{rr} 1 & 24 \\ 24 & 26 \end{array}\right) $ |
| $C_{10}.S_4$ | 40A-1 | $40$ | $6$ | $C_{40}$ | 20A-1 | 40A-3 | 8A1 | $ \left(\begin{array}{rr} 4 & 3 \\ 3 & 11 \end{array}\right) $ |
| $C_{10}.S_4$ | 40A3 | $40$ | $6$ | $C_{40}$ | 20A3 | 40A-1 | 8A3 | $ \left(\begin{array}{rr} 4 & 27 \\ 7 & 28 \end{array}\right) $ |
| $C_{10}.S_4$ | 40A-3 | $40$ | $6$ | $C_{40}$ | 20A-3 | 40A1 | 8A3 | $ \left(\begin{array}{rr} 8 & 23 \\ 14 & 25 \end{array}\right) $ |
| $C_{10}.S_4$ | 40A7 | $40$ | $6$ | $C_{40}$ | 20A-3 | 40A11 | 8A1 | $ \left(\begin{array}{rr} 6 & 23 \\ 14 & 23 \end{array}\right) $ |
| $C_{10}.S_4$ | 40A-7 | $40$ | $6$ | $C_{40}$ | 20A3 | 40A-11 | 8A1 | $ \left(\begin{array}{rr} 3 & 27 \\ 7 & 27 \end{array}\right) $ |
| $C_{10}.S_4$ | 40A11 | $40$ | $6$ | $C_{40}$ | 20A1 | 40A-7 | 8A3 | $ \left(\begin{array}{rr} 5 & 24 \\ 24 & 30 \end{array}\right) $ |
| $C_{10}.S_4$ | 40A-11 | $40$ | $6$ | $C_{40}$ | 20A-1 | 40A7 | 8A3 | $ \left(\begin{array}{rr} 2 & 29 \\ 19 & 14 \end{array}\right) $ |