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Elements of the group are displayed as matrices in $\GL_{2}(\F_{53})$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 13P | ||||||
| $C_{52}\wr C_2$ | 1A | $1$ | $1$ | $C_{52}\wr C_2$ | 1A | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 2A | $2$ | $1$ | $C_{52}\wr C_2$ | 1A | 2A | $ \left(\begin{array}{rr} 52 & 0 \\ 0 & 52 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 2B | $2$ | $2$ | $C_{52}^2$ | 1A | 2B | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 52 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 2C | $2$ | $52$ | $C_2\times C_{52}$ | 1A | 2C | $ \left(\begin{array}{rr} 0 & 49 \\ 13 & 0 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 4A1 | $4$ | $1$ | $C_{52}\wr C_2$ | 2A | 4A-1 | $ \left(\begin{array}{rr} 30 & 0 \\ 0 & 30 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 4A-1 | $4$ | $1$ | $C_{52}\wr C_2$ | 2A | 4A1 | $ \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 4B | $4$ | $2$ | $C_{52}^2$ | 2A | 4B | $ \left(\begin{array}{rr} 30 & 0 \\ 0 & 23 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 4C1 | $4$ | $2$ | $C_{52}^2$ | 2B | 4C-1 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 30 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 4C-1 | $4$ | $2$ | $C_{52}^2$ | 2B | 4C1 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 23 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 4D1 | $4$ | $2$ | $C_{52}^2$ | 2B | 4D-1 | $ \left(\begin{array}{rr} 30 & 0 \\ 0 & 52 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 4D-1 | $4$ | $2$ | $C_{52}^2$ | 2B | 4D1 | $ \left(\begin{array}{rr} 23 & 0 \\ 0 & 52 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 4E | $4$ | $52$ | $C_2\times C_{52}$ | 2A | 4E | $ \left(\begin{array}{rr} 0 & 15 \\ 7 & 0 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 8A1 | $8$ | $52$ | $C_{104}$ | 4A1 | 8A-1 | $ \left(\begin{array}{rr} 0 & 11 \\ 22 & 0 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 8A-1 | $8$ | $52$ | $C_{104}$ | 4A-1 | 8A1 | $ \left(\begin{array}{rr} 0 & 41 \\ 29 & 0 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13A1 | $13$ | $1$ | $C_{52}\wr C_2$ | 13A2 | 13A3 | $ \left(\begin{array}{rr} 10 & 0 \\ 0 & 10 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13A-1 | $13$ | $1$ | $C_{52}\wr C_2$ | 13A-2 | 13A-3 | $ \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13A2 | $13$ | $1$ | $C_{52}\wr C_2$ | 13A4 | 13A6 | $ \left(\begin{array}{rr} 47 & 0 \\ 0 & 47 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13A-2 | $13$ | $1$ | $C_{52}\wr C_2$ | 13A-4 | 13A-6 | $ \left(\begin{array}{rr} 44 & 0 \\ 0 & 44 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13A3 | $13$ | $1$ | $C_{52}\wr C_2$ | 13A6 | 13A-4 | $ \left(\begin{array}{rr} 46 & 0 \\ 0 & 46 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13A-3 | $13$ | $1$ | $C_{52}\wr C_2$ | 13A-6 | 13A4 | $ \left(\begin{array}{rr} 15 & 0 \\ 0 & 15 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13A4 | $13$ | $1$ | $C_{52}\wr C_2$ | 13A-5 | 13A-1 | $ \left(\begin{array}{rr} 36 & 0 \\ 0 & 36 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13A-4 | $13$ | $1$ | $C_{52}\wr C_2$ | 13A5 | 13A1 | $ \left(\begin{array}{rr} 28 & 0 \\ 0 & 28 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13A5 | $13$ | $1$ | $C_{52}\wr C_2$ | 13A-3 | 13A2 | $ \left(\begin{array}{rr} 42 & 0 \\ 0 & 42 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13A-5 | $13$ | $1$ | $C_{52}\wr C_2$ | 13A3 | 13A-2 | $ \left(\begin{array}{rr} 24 & 0 \\ 0 & 24 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13A6 | $13$ | $1$ | $C_{52}\wr C_2$ | 13A-1 | 13A5 | $ \left(\begin{array}{rr} 49 & 0 \\ 0 & 49 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13A-6 | $13$ | $1$ | $C_{52}\wr C_2$ | 13A1 | 13A-5 | $ \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13B1 | $13$ | $2$ | $C_{52}^2$ | 13B2 | 13B3 | $ \left(\begin{array}{rr} 16 & 0 \\ 0 & 10 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13B2 | $13$ | $2$ | $C_{52}^2$ | 13B4 | 13B6 | $ \left(\begin{array}{rr} 44 & 0 \\ 0 & 47 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13B3 | $13$ | $2$ | $C_{52}^2$ | 13B6 | 13B4 | $ \left(\begin{array}{rr} 15 & 0 \\ 0 & 46 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13B4 | $13$ | $2$ | $C_{52}^2$ | 13B5 | 13B1 | $ \left(\begin{array}{rr} 28 & 0 \\ 0 & 36 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13B5 | $13$ | $2$ | $C_{52}^2$ | 13B3 | 13B2 | $ \left(\begin{array}{rr} 24 & 0 \\ 0 & 42 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13B6 | $13$ | $2$ | $C_{52}^2$ | 13B1 | 13B5 | $ \left(\begin{array}{rr} 13 & 0 \\ 0 & 49 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13C1 | $13$ | $2$ | $C_{52}^2$ | 13C2 | 13C3 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 16 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13C-1 | $13$ | $2$ | $C_{52}^2$ | 13C-2 | 13C-3 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 10 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13C2 | $13$ | $2$ | $C_{52}^2$ | 13C4 | 13C6 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 44 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13C-2 | $13$ | $2$ | $C_{52}^2$ | 13C-4 | 13C-6 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 47 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13C3 | $13$ | $2$ | $C_{52}^2$ | 13C6 | 13C-4 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 15 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13C-3 | $13$ | $2$ | $C_{52}^2$ | 13C-6 | 13C4 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 46 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13C4 | $13$ | $2$ | $C_{52}^2$ | 13C-5 | 13C-1 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 28 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13C-4 | $13$ | $2$ | $C_{52}^2$ | 13C5 | 13C1 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 36 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13C5 | $13$ | $2$ | $C_{52}^2$ | 13C-3 | 13C2 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 24 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13C-5 | $13$ | $2$ | $C_{52}^2$ | 13C3 | 13C-2 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 42 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13C6 | $13$ | $2$ | $C_{52}^2$ | 13C-1 | 13C5 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 13 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13C-6 | $13$ | $2$ | $C_{52}^2$ | 13C1 | 13C-5 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 49 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13D1 | $13$ | $2$ | $C_{52}^2$ | 13D2 | 13D3 | $ \left(\begin{array}{rr} 16 & 0 \\ 0 & 28 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13D-1 | $13$ | $2$ | $C_{52}^2$ | 13D-2 | 13D-3 | $ \left(\begin{array}{rr} 10 & 0 \\ 0 & 36 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13D2 | $13$ | $2$ | $C_{52}^2$ | 13D4 | 13D6 | $ \left(\begin{array}{rr} 44 & 0 \\ 0 & 42 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13D-2 | $13$ | $2$ | $C_{52}^2$ | 13D-4 | 13D-6 | $ \left(\begin{array}{rr} 47 & 0 \\ 0 & 24 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13D3 | $13$ | $2$ | $C_{52}^2$ | 13D6 | 13D-4 | $ \left(\begin{array}{rr} 15 & 0 \\ 0 & 10 \end{array}\right) $ |
| $C_{52}\wr C_2$ | 13D-3 | $13$ | $2$ | $C_{52}^2$ | 13D-6 | 13D4 | $ \left(\begin{array}{rr} 46 & 0 \\ 0 & 16 \end{array}\right) $ |