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Elements of the group are displayed as matrices in $\GL_{2}(\Z/{20}\Z)$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | ||
|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 5P | ||||||
| $C_2^2\times F_5\times S_4$ | 1A | $1$ | $1$ | $C_2^2\times F_5\times S_4$ | 1A | 1A | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2A | $2$ | $1$ | $C_2^2\times F_5\times S_4$ | 1A | 2A | 2A | $ \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2B | $2$ | $1$ | $C_2^2\times F_5\times S_4$ | 1A | 2B | 2B | $ \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2C | $2$ | $1$ | $C_2^2\times F_5\times S_4$ | 1A | 2C | 2C | $ \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2D | $2$ | $3$ | $D_{10}.C_2^5$ | 1A | 2D | 2D | $ \left(\begin{array}{rr} 11 & 0 \\ 10 & 11 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2E | $2$ | $3$ | $D_{10}.C_2^5$ | 1A | 2E | 2E | $ \left(\begin{array}{rr} 19 & 0 \\ 10 & 19 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2F | $2$ | $3$ | $D_{10}.C_2^5$ | 1A | 2F | 2F | $ \left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2G | $2$ | $3$ | $D_{10}.C_2^5$ | 1A | 2G | 2G | $ \left(\begin{array}{rr} 9 & 0 \\ 10 & 9 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2H | $2$ | $5$ | $C_2^5.D_6$ | 1A | 2H | 2H | $ \left(\begin{array}{rr} 9 & 4 \\ 0 & 1 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2I | $2$ | $5$ | $C_2^5.D_6$ | 1A | 2I | 2I | $ \left(\begin{array}{rr} 1 & 16 \\ 0 & 9 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2J | $2$ | $5$ | $C_2^5.D_6$ | 1A | 2J | 2J | $ \left(\begin{array}{rr} 19 & 4 \\ 0 & 11 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2K | $2$ | $5$ | $C_2^5.D_6$ | 1A | 2K | 2K | $ \left(\begin{array}{rr} 11 & 16 \\ 0 & 19 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2L | $2$ | $6$ | $C_2^4\times F_5$ | 1A | 2L | 2L | $ \left(\begin{array}{rr} 11 & 5 \\ 0 & 1 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2M | $2$ | $6$ | $C_2^4\times F_5$ | 1A | 2M | 2M | $ \left(\begin{array}{rr} 19 & 5 \\ 0 & 9 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2N | $2$ | $6$ | $C_2^4\times F_5$ | 1A | 2N | 2N | $ \left(\begin{array}{rr} 1 & 15 \\ 0 & 11 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2O | $2$ | $6$ | $C_2^4\times F_5$ | 1A | 2O | 2O | $ \left(\begin{array}{rr} 9 & 15 \\ 0 & 19 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2P | $2$ | $15$ | $C_4^2:C_2^3$ | 1A | 2P | 2P | $ \left(\begin{array}{rr} 11 & 6 \\ 0 & 19 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2Q | $2$ | $15$ | $C_4^2:C_2^3$ | 1A | 2Q | 2Q | $ \left(\begin{array}{rr} 9 & 14 \\ 0 & 1 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2R | $2$ | $15$ | $C_4^2:C_2^3$ | 1A | 2R | 2R | $ \left(\begin{array}{rr} 1 & 6 \\ 0 & 9 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2S | $2$ | $15$ | $C_4^2:C_2^3$ | 1A | 2S | 2S | $ \left(\begin{array}{rr} 9 & 14 \\ 10 & 1 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2T | $2$ | $30$ | $C_2^4\times C_4$ | 1A | 2T | 2T | $ \left(\begin{array}{rr} 9 & 4 \\ 5 & 11 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2U | $2$ | $30$ | $C_2^4\times C_4$ | 1A | 2U | 2U | $ \left(\begin{array}{rr} 1 & 16 \\ 5 & 19 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2V | $2$ | $30$ | $C_2^4\times C_4$ | 1A | 2V | 2V | $ \left(\begin{array}{rr} 19 & 4 \\ 15 & 1 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 2W | $2$ | $30$ | $C_2^4\times C_4$ | 1A | 2W | 2W | $ \left(\begin{array}{rr} 11 & 16 \\ 15 & 9 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 3A | $3$ | $8$ | $D_{10}:C_{12}$ | 3A | 1A | 3A | $ \left(\begin{array}{rr} 11 & 5 \\ 15 & 16 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4A1 | $4$ | $5$ | $C_2^5.D_6$ | 2H | 4A-1 | 4A1 | $ \left(\begin{array}{rr} 17 & 8 \\ 0 & 1 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4A-1 | $4$ | $5$ | $C_2^5.D_6$ | 2H | 4A1 | 4A-1 | $ \left(\begin{array}{rr} 13 & 16 \\ 0 & 1 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4B1 | $4$ | $5$ | $C_2^5.D_6$ | 2H | 4B-1 | 4B1 | $ \left(\begin{array}{rr} 13 & 12 \\ 0 & 9 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4B-1 | $4$ | $5$ | $C_2^5.D_6$ | 2H | 4B1 | 4B-1 | $ \left(\begin{array}{rr} 17 & 4 \\ 0 & 9 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4C1 | $4$ | $5$ | $C_2^5.D_6$ | 2H | 4C-1 | 4C1 | $ \left(\begin{array}{rr} 7 & 8 \\ 0 & 11 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4C-1 | $4$ | $5$ | $C_2^5.D_6$ | 2H | 4C1 | 4C-1 | $ \left(\begin{array}{rr} 3 & 16 \\ 0 & 11 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4D1 | $4$ | $5$ | $C_2^5.D_6$ | 2H | 4D-1 | 4D1 | $ \left(\begin{array}{rr} 3 & 12 \\ 0 & 19 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4D-1 | $4$ | $5$ | $C_2^5.D_6$ | 2H | 4D1 | 4D-1 | $ \left(\begin{array}{rr} 7 & 4 \\ 0 & 19 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4E | $4$ | $6$ | $D_{10}:C_4^2$ | 2D | 4E | 4E | $ \left(\begin{array}{rr} 11 & 15 \\ 10 & 11 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4F | $4$ | $6$ | $D_{10}:C_4^2$ | 2D | 4F | 4F | $ \left(\begin{array}{rr} 19 & 15 \\ 10 & 19 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4G | $4$ | $6$ | $D_{10}:C_4^2$ | 2D | 4G | 4G | $ \left(\begin{array}{rr} 1 & 5 \\ 10 & 1 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4H | $4$ | $6$ | $D_{10}:C_4^2$ | 2D | 4H | 4H | $ \left(\begin{array}{rr} 9 & 5 \\ 10 & 9 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4I1 | $4$ | $15$ | $C_4^2:C_2^3$ | 2H | 4I-1 | 4I1 | $ \left(\begin{array}{rr} 7 & 18 \\ 0 & 11 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4I-1 | $4$ | $15$ | $C_4^2:C_2^3$ | 2H | 4I1 | 4I-1 | $ \left(\begin{array}{rr} 3 & 6 \\ 0 & 11 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4J1 | $4$ | $15$ | $C_4^2:C_2^3$ | 2H | 4J-1 | 4J1 | $ \left(\begin{array}{rr} 3 & 2 \\ 0 & 19 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4J-1 | $4$ | $15$ | $C_4^2:C_2^3$ | 2H | 4J1 | 4J-1 | $ \left(\begin{array}{rr} 7 & 14 \\ 0 & 19 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4K1 | $4$ | $15$ | $C_4^2:C_2^3$ | 2H | 4K-1 | 4K1 | $ \left(\begin{array}{rr} 17 & 18 \\ 0 & 1 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4K-1 | $4$ | $15$ | $C_4^2:C_2^3$ | 2H | 4K1 | 4K-1 | $ \left(\begin{array}{rr} 13 & 6 \\ 0 & 1 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4L1 | $4$ | $15$ | $C_4^2:C_2^3$ | 2H | 4L-1 | 4L1 | $ \left(\begin{array}{rr} 13 & 2 \\ 0 & 9 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4L-1 | $4$ | $15$ | $C_4^2:C_2^3$ | 2H | 4L1 | 4L-1 | $ \left(\begin{array}{rr} 17 & 14 \\ 0 & 9 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4M | $4$ | $30$ | $C_2^2\times C_4^2$ | 2D | 4M | 4M | $ \left(\begin{array}{rr} 19 & 14 \\ 15 & 11 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4N | $4$ | $30$ | $C_2^2\times C_4^2$ | 2D | 4N | 4N | $ \left(\begin{array}{rr} 11 & 6 \\ 15 & 19 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4O | $4$ | $30$ | $C_2^2\times C_4^2$ | 2D | 4O | 4O | $ \left(\begin{array}{rr} 9 & 14 \\ 5 & 1 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4P | $4$ | $30$ | $C_2^2\times C_4^2$ | 2D | 4P | 4P | $ \left(\begin{array}{rr} 1 & 6 \\ 5 & 9 \end{array}\right) $ |
| $C_2^2\times F_5\times S_4$ | 4Q1 | $4$ | $30$ | $C_2^4\times C_4$ | 2H | 4Q-1 | 4Q1 | $ \left(\begin{array}{rr} 8 & 11 \\ 15 & 16 \end{array}\right) $ |