Refine search
Elements of the group are displayed as matrices in $\GL_{4}(\F_{4})$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 3P | ||||||
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 1A | $1$ | $1$ | $C_2^3:\GL(2,\mathbb{Z}/4)$ | 1A | 1A | $\left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2A | $2$ | $1$ | $C_2^3:\GL(2,\mathbb{Z}/4)$ | 1A | 2A | $\left(\begin{array}{llll}1 & 0 & 0 & 1 \\ 1 & \alpha^{2} & \alpha^{2} & 0 \\ \alpha^{2} & 1 & \alpha^{2} & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2B | $2$ | $1$ | $C_2^3:\GL(2,\mathbb{Z}/4)$ | 1A | 2B | $\left(\begin{array}{llll}1 & 0 & 0 & \alpha \\ \alpha & \alpha & 1 & 0 \\ 1 & \alpha & \alpha & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2C | $2$ | $1$ | $C_2^3:\GL(2,\mathbb{Z}/4)$ | 1A | 2C | $\left(\begin{array}{llll}1 & 0 & 0 & \alpha^{2} \\ \alpha^{2} & 0 & \alpha & 0 \\ \alpha & \alpha^{2} & 0 & 1 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2D | $2$ | $2$ | $C_2^5:A_4$ | 1A | 2D | $\left(\begin{array}{llll}1 & 0 & 0 & \alpha^{2} \\ \alpha & \alpha & 1 & 0 \\ 1 & \alpha & \alpha & 1 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2E | $2$ | $2$ | $C_2^5:A_4$ | 1A | 2E | $\left(\begin{array}{llll}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2F | $2$ | $3$ | $C_2^5:D_4$ | 1A | 2F | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & \alpha^{2} \\ \alpha^{2} & 0 & \alpha & \alpha \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2G | $2$ | $3$ | $C_2^5:D_4$ | 1A | 2G | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & \alpha^{2} \\ \alpha^{2} & 0 & \alpha & 1 \\ 0 & 0 & 1 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2H | $2$ | $3$ | $C_2^5:D_4$ | 1A | 2H | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & \alpha^{2} \\ \alpha^{2} & 0 & \alpha & \alpha^{2} \\ 0 & 0 & 1 & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2I | $2$ | $3$ | $C_2^5:D_4$ | 1A | 2I | $\left(\begin{array}{llll}\alpha & 1 & \alpha & \alpha \\ \alpha & \alpha & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2J | $2$ | $3$ | $C_2^5:D_4$ | 1A | 2J | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & \alpha \\ \alpha & \alpha & 1 & 1 \\ \alpha^{2} & 1 & \alpha^{2} & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2K | $2$ | $3$ | $C_2^5:D_4$ | 1A | 2K | $\left(\begin{array}{llll}\alpha & 1 & \alpha & \alpha \\ \alpha & \alpha & 1 & \alpha^{2} \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2L | $2$ | $3$ | $C_2^5:D_4$ | 1A | 2L | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & 0 \\ 0 & 1 & 0 & 1 \\ \alpha & \alpha^{2} & 0 & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2M | $2$ | $3$ | $C_2^5:D_4$ | 1A | 2M | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & 0 \\ 0 & 1 & 0 & \alpha \\ \alpha & \alpha^{2} & 0 & 1 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2N | $2$ | $3$ | $C_2^5:D_4$ | 1A | 2N | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & 0 \\ 0 & 1 & 0 & \alpha^{2} \\ \alpha & \alpha^{2} & 0 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2O | $2$ | $3$ | $C_2^5:D_4$ | 1A | 2O | $\left(\begin{array}{llll}\alpha & 1 & \alpha & 1 \\ 1 & \alpha^{2} & \alpha^{2} & \alpha^{2} \\ \alpha & \alpha^{2} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2P | $2$ | $3$ | $C_2^5:D_4$ | 1A | 2P | $\left(\begin{array}{llll}\alpha^{2} & \alpha^{2} & 1 & 1 \\ 1 & \alpha^{2} & \alpha^{2} & \alpha^{2} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2Q | $2$ | $3$ | $C_2^5:D_4$ | 1A | 2Q | $\left(\begin{array}{llll}\alpha & 1 & \alpha & 1 \\ 1 & \alpha^{2} & \alpha^{2} & 1 \\ \alpha & \alpha^{2} & 0 & 1 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2R | $2$ | $6$ | $C_2^7$ | 1A | 2R | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & 0 \\ 0 & 1 & 0 & 0 \\ \alpha & \alpha^{2} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2S | $2$ | $6$ | $C_2^7$ | 1A | 2S | $\left(\begin{array}{llll}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & \alpha^{2} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2T | $2$ | $6$ | $C_2^7$ | 1A | 2T | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & 1 \\ 0 & 1 & 0 & 0 \\ \alpha & \alpha^{2} & 0 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2U | $2$ | $6$ | $C_2^7$ | 1A | 2U | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & 1 \\ 0 & 1 & 0 & \alpha^{2} \\ \alpha & \alpha^{2} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2V | $2$ | $6$ | $C_2^7$ | 1A | 2V | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & \alpha \\ \alpha^{2} & 0 & \alpha & 0 \\ 0 & 0 & 1 & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2W | $2$ | $6$ | $C_2^7$ | 1A | 2W | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & \alpha \\ \alpha^{2} & 0 & \alpha & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2X | $2$ | $6$ | $C_2^7$ | 1A | 2X | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & \alpha^{2} \\ \alpha^{2} & 0 & \alpha & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2Y | $2$ | $6$ | $C_2^7$ | 1A | 2Y | $\left(\begin{array}{llll}1 & 0 & 0 & \alpha \\ \alpha^{2} & 0 & \alpha & 1 \\ \alpha & \alpha^{2} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2Z | $2$ | $6$ | $C_2^7$ | 1A | 2Z | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & 1 \\ 0 & 1 & 0 & 1 \\ \alpha & \alpha^{2} & 0 & 1 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2AA | $2$ | $6$ | $C_2^7$ | 1A | 2AA | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & 1 \\ 0 & 1 & 0 & \alpha \\ \alpha & \alpha^{2} & 0 & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2AB | $2$ | $6$ | $C_2^7$ | 1A | 2AB | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & \alpha \\ \alpha^{2} & 0 & \alpha & \alpha \\ 0 & 0 & 1 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2AC | $2$ | $6$ | $C_2^7$ | 1A | 2AC | $\left(\begin{array}{llll}0 & \alpha & \alpha^{2} & \alpha \\ \alpha^{2} & 0 & \alpha & \alpha^{2} \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2AD | $2$ | $6$ | $C_2^7$ | 1A | 2AD | $\left(\begin{array}{llll}1 & 0 & 0 & 1 \\ 1 & \alpha^{2} & \alpha^{2} & \alpha^{2} \\ \alpha^{2} & 1 & \alpha^{2} & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2AE | $2$ | $6$ | $C_2^7$ | 1A | 2AE | $\left(\begin{array}{llll}1 & 0 & 0 & \alpha \\ \alpha & \alpha & 1 & 1 \\ 1 & \alpha & \alpha & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2AF | $2$ | $24$ | $C_2^5$ | 1A | 2AF | $\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \alpha^{2} & \alpha & 0 & \alpha \\ 1 & \alpha & \alpha^{2} & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 2AG | $2$ | $24$ | $C_2^5$ | 1A | 2AG | $\left(\begin{array}{llll}1 & 0 & \alpha^{2} & 0 \\ \alpha^{2} & 0 & 0 & \alpha \\ 0 & 0 & 1 & 0 \\ \alpha & \alpha^{2} & 1 & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 3A | $3$ | $32$ | $C_2^2\times C_6$ | 3A | 1A | $\left(\begin{array}{llll}0 & \alpha^{2} & 1 & 0 \\ 0 & \alpha^{2} & 0 & \alpha^{2} \\ 1 & 1 & 1 & \alpha \\ 0 & 0 & 0 & \alpha \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4A | $4$ | $24$ | $C_2^3\times C_4$ | 2L | 4A | $\left(\begin{array}{llll}0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ \alpha^{2} & \alpha & 0 & 0 \\ 1 & \alpha & \alpha^{2} & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4B | $4$ | $24$ | $C_2^3\times C_4$ | 2M | 4B | $\left(\begin{array}{llll}0 & \alpha & 0 & 0 \\ 0 & 1 & \alpha & 0 \\ 0 & 0 & 1 & 1 \\ \alpha & \alpha^{2} & 1 & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4C | $4$ | $24$ | $C_2^3\times C_4$ | 2N | 4C | $\left(\begin{array}{llll}0 & \alpha^{2} & 0 & \alpha \\ \alpha & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ \alpha^{2} & 1 & \alpha & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4D | $4$ | $24$ | $C_2^3\times C_4$ | 2A | 4D | $\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ \alpha^{2} & \alpha & 0 & 1 \\ 1 & \alpha & \alpha^{2} & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4E | $4$ | $24$ | $C_2^3\times C_4$ | 2O | 4E | $\left(\begin{array}{llll}0 & 1 & 0 & \alpha^{2} \\ 1 & 0 & 0 & 1 \\ \alpha^{2} & \alpha & 0 & 0 \\ 1 & \alpha & \alpha^{2} & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4F | $4$ | $24$ | $C_2^3\times C_4$ | 2P | 4F | $\left(\begin{array}{llll}0 & \alpha & 0 & \alpha \\ 0 & 1 & \alpha & \alpha^{2} \\ 0 & 0 & 1 & 0 \\ \alpha & \alpha^{2} & 1 & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4G | $4$ | $24$ | $C_2^3\times C_4$ | 2Q | 4G | $\left(\begin{array}{llll}0 & \alpha & 0 & \alpha \\ \alpha^{2} & 0 & 0 & \alpha^{2} \\ \alpha & \alpha^{2} & 0 & 0 \\ \alpha & \alpha^{2} & 1 & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4H | $4$ | $24$ | $C_2^3\times C_4$ | 2A | 4H | $\left(\begin{array}{llll}1 & 0 & \alpha^{2} & 0 \\ \alpha^{2} & 0 & 0 & 1 \\ 0 & 0 & 1 & \alpha \\ \alpha & \alpha^{2} & 1 & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4I | $4$ | $24$ | $C_2^3\times C_4$ | 2P | 4I | $\left(\begin{array}{llll}1 & 0 & \alpha^{2} & 1 \\ \alpha^{2} & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \alpha & \alpha^{2} & 1 & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4J | $4$ | $24$ | $C_2^3\times C_4$ | 2M | 4J | $\left(\begin{array}{llll}1 & 0 & \alpha^{2} & 0 \\ 0 & 1 & \alpha & \alpha \\ \alpha & \alpha^{2} & 0 & 0 \\ \alpha & \alpha^{2} & 1 & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4K | $4$ | $24$ | $C_2^3\times C_4$ | 2L | 4K | $\left(\begin{array}{llll}1 & 0 & \alpha^{2} & 1 \\ \alpha^{2} & 0 & 0 & \alpha \\ 0 & 0 & 1 & \alpha \\ \alpha & \alpha^{2} & 1 & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4L | $4$ | $24$ | $C_2^3\times C_4$ | 2Q | 4L | $\left(\begin{array}{llll}1 & 0 & \alpha^{2} & \alpha \\ \alpha^{2} & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ \alpha & \alpha^{2} & 1 & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4M | $4$ | $24$ | $C_2^3\times C_4$ | 2N | 4M | $\left(\begin{array}{llll}1 & 0 & \alpha^{2} & \alpha \\ \alpha^{2} & 0 & 0 & \alpha \\ 0 & 0 & 1 & \alpha^{2} \\ \alpha & \alpha^{2} & 1 & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 4N | $4$ | $24$ | $C_2^3\times C_4$ | 2O | 4N | $\left(\begin{array}{llll}1 & 0 & \alpha^{2} & \alpha^{2} \\ \alpha^{2} & 0 & 0 & 1 \\ 0 & 0 & 1 & \alpha^{2} \\ \alpha & \alpha^{2} & 1 & 0 \\ \end{array}\right)$ |
| $C_2^3:\GL(2,\mathbb{Z}/4)$ | 6A | $6$ | $32$ | $C_2^2\times C_6$ | 3A | 2B | $\left(\begin{array}{llll}\alpha^{2} & 0 & 0 & 1 \\ \alpha^{2} & \alpha^{2} & \alpha & 0 \\ 0 & \alpha^{2} & \alpha^{2} & \alpha \\ 0 & 0 & 0 & \alpha^{2} \\ \end{array}\right)$ |