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Elements 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) $
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