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Elements of the group are displayed as matrices in $\GL_{2}(\F_{71})$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | ||
|---|---|---|---|---|---|---|---|---|
| 2P | 5P | 7P | ||||||
| $C_{70}\wr C_2$ | 1A | $1$ | $1$ | $C_{70}\wr C_2$ | 1A | 1A | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 2A | $2$ | $1$ | $C_{70}\wr C_2$ | 1A | 2A | 2A | $ \left(\begin{array}{rr} 70 & 0 \\ 0 & 70 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 2B | $2$ | $2$ | $C_{70}^2$ | 1A | 2B | 2B | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 70 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 2C | $2$ | $70$ | $C_2\times C_{70}$ | 1A | 2C | 2C | $ \left(\begin{array}{rr} 0 & 6 \\ 12 & 0 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 4A | $4$ | $70$ | $C_{140}$ | 2A | 4A | 4A | $ \left(\begin{array}{rr} 0 & 62 \\ 8 & 0 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5A1 | $5$ | $1$ | $C_{70}\wr C_2$ | 5A2 | 5A-2 | 1A | $ \left(\begin{array}{rr} 54 & 0 \\ 0 & 54 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5A-1 | $5$ | $1$ | $C_{70}\wr C_2$ | 5A-2 | 5A2 | 1A | $ \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5A2 | $5$ | $1$ | $C_{70}\wr C_2$ | 5A-1 | 5A1 | 1A | $ \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5A-2 | $5$ | $1$ | $C_{70}\wr C_2$ | 5A1 | 5A-1 | 1A | $ \left(\begin{array}{rr} 57 & 0 \\ 0 & 57 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5B1 | $5$ | $2$ | $C_{70}^2$ | 5B2 | 5B2 | 1A | $ \left(\begin{array}{rr} 54 & 0 \\ 0 & 25 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5B2 | $5$ | $2$ | $C_{70}^2$ | 5B1 | 5B1 | 1A | $ \left(\begin{array}{rr} 5 & 0 \\ 0 & 57 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5C1 | $5$ | $2$ | $C_{70}^2$ | 5C2 | 5C-2 | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 54 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5C-1 | $5$ | $2$ | $C_{70}^2$ | 5C-2 | 5C2 | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 25 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5C2 | $5$ | $2$ | $C_{70}^2$ | 5C-1 | 5C1 | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 5 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5C-2 | $5$ | $2$ | $C_{70}^2$ | 5C1 | 5C-1 | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 57 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5D1 | $5$ | $2$ | $C_{70}^2$ | 5D2 | 5D-2 | 1A | $ \left(\begin{array}{rr} 54 & 0 \\ 0 & 5 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5D-1 | $5$ | $2$ | $C_{70}^2$ | 5D-2 | 5D2 | 1A | $ \left(\begin{array}{rr} 25 & 0 \\ 0 & 57 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5D2 | $5$ | $2$ | $C_{70}^2$ | 5D-1 | 5D1 | 1A | $ \left(\begin{array}{rr} 5 & 0 \\ 0 & 25 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 5D-2 | $5$ | $2$ | $C_{70}^2$ | 5D1 | 5D-1 | 1A | $ \left(\begin{array}{rr} 57 & 0 \\ 0 & 54 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7A1 | $7$ | $1$ | $C_{70}\wr C_2$ | 7A2 | 7A3 | 7A-2 | $ \left(\begin{array}{rr} 45 & 0 \\ 0 & 45 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7A-1 | $7$ | $1$ | $C_{70}\wr C_2$ | 7A-2 | 7A-3 | 7A2 | $ \left(\begin{array}{rr} 30 & 0 \\ 0 & 30 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7A2 | $7$ | $1$ | $C_{70}\wr C_2$ | 7A-3 | 7A-1 | 7A3 | $ \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7A-2 | $7$ | $1$ | $C_{70}\wr C_2$ | 7A3 | 7A1 | 7A-3 | $ \left(\begin{array}{rr} 48 & 0 \\ 0 & 48 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7A3 | $7$ | $1$ | $C_{70}\wr C_2$ | 7A-1 | 7A2 | 7A1 | $ \left(\begin{array}{rr} 32 & 0 \\ 0 & 32 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7A-3 | $7$ | $1$ | $C_{70}\wr C_2$ | 7A1 | 7A-2 | 7A-1 | $ \left(\begin{array}{rr} 20 & 0 \\ 0 & 20 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7B1 | $7$ | $2$ | $C_{70}^2$ | 7B2 | 7B3 | 7B2 | $ \left(\begin{array}{rr} 45 & 0 \\ 0 & 30 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7B2 | $7$ | $2$ | $C_{70}^2$ | 7B3 | 7B1 | 7B3 | $ \left(\begin{array}{rr} 37 & 0 \\ 0 & 48 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7B3 | $7$ | $2$ | $C_{70}^2$ | 7B1 | 7B2 | 7B1 | $ \left(\begin{array}{rr} 32 & 0 \\ 0 & 20 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7C1 | $7$ | $2$ | $C_{70}^2$ | 7C2 | 7C3 | 7C-2 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 45 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7C-1 | $7$ | $2$ | $C_{70}^2$ | 7C-2 | 7C-3 | 7C2 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 30 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7C2 | $7$ | $2$ | $C_{70}^2$ | 7C-3 | 7C-1 | 7C3 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 37 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7C-2 | $7$ | $2$ | $C_{70}^2$ | 7C3 | 7C1 | 7C-3 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 48 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7C3 | $7$ | $2$ | $C_{70}^2$ | 7C-1 | 7C2 | 7C1 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 32 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7C-3 | $7$ | $2$ | $C_{70}^2$ | 7C1 | 7C-2 | 7C-1 | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 20 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7D1 | $7$ | $2$ | $C_{70}^2$ | 7D2 | 7D3 | 7D-2 | $ \left(\begin{array}{rr} 30 & 0 \\ 0 & 32 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7D-1 | $7$ | $2$ | $C_{70}^2$ | 7D-2 | 7D-3 | 7D2 | $ \left(\begin{array}{rr} 45 & 0 \\ 0 & 20 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7D2 | $7$ | $2$ | $C_{70}^2$ | 7D-3 | 7D-1 | 7D3 | $ \left(\begin{array}{rr} 48 & 0 \\ 0 & 30 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7D-2 | $7$ | $2$ | $C_{70}^2$ | 7D3 | 7D1 | 7D-3 | $ \left(\begin{array}{rr} 37 & 0 \\ 0 & 45 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7D3 | $7$ | $2$ | $C_{70}^2$ | 7D-1 | 7D2 | 7D1 | $ \left(\begin{array}{rr} 20 & 0 \\ 0 & 37 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7D-3 | $7$ | $2$ | $C_{70}^2$ | 7D1 | 7D-2 | 7D-1 | $ \left(\begin{array}{rr} 32 & 0 \\ 0 & 48 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7E1 | $7$ | $2$ | $C_{70}^2$ | 7E2 | 7E3 | 7E-2 | $ \left(\begin{array}{rr} 30 & 0 \\ 0 & 20 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7E-1 | $7$ | $2$ | $C_{70}^2$ | 7E-2 | 7E-3 | 7E2 | $ \left(\begin{array}{rr} 45 & 0 \\ 0 & 32 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7E2 | $7$ | $2$ | $C_{70}^2$ | 7E-3 | 7E-1 | 7E3 | $ \left(\begin{array}{rr} 48 & 0 \\ 0 & 45 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7E-2 | $7$ | $2$ | $C_{70}^2$ | 7E3 | 7E1 | 7E-3 | $ \left(\begin{array}{rr} 37 & 0 \\ 0 & 30 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7E3 | $7$ | $2$ | $C_{70}^2$ | 7E-1 | 7E2 | 7E1 | $ \left(\begin{array}{rr} 20 & 0 \\ 0 & 48 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 7E-3 | $7$ | $2$ | $C_{70}^2$ | 7E1 | 7E-2 | 7E-1 | $ \left(\begin{array}{rr} 32 & 0 \\ 0 & 37 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 10A1 | $10$ | $1$ | $C_{70}\wr C_2$ | 5A1 | 10A3 | 2A | $ \left(\begin{array}{rr} 14 & 0 \\ 0 & 14 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 10A-1 | $10$ | $1$ | $C_{70}\wr C_2$ | 5A-1 | 10A-3 | 2A | $ \left(\begin{array}{rr} 66 & 0 \\ 0 & 66 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 10A3 | $10$ | $1$ | $C_{70}\wr C_2$ | 5A-2 | 10A-1 | 2A | $ \left(\begin{array}{rr} 46 & 0 \\ 0 & 46 \end{array}\right) $ |
| $C_{70}\wr C_2$ | 10A-3 | $10$ | $1$ | $C_{70}\wr C_2$ | 5A2 | 10A1 | 2A | $ \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right) $ |