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Elements of the group are displayed as matrices in $\GL_{2}(\F_{243})$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 3P | ||||||
| $C_3^4:S_3$ | 1A | $1$ | $1$ | $C_3^4:S_3$ | 1A | 1A | $\left(\begin{array}{ll}1 & 0 \\ 0 & 1 \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 2A | $2$ | $243$ | $C_2$ | 1A | 2A | $\left(\begin{array}{ll}\alpha^{89} & \alpha^{214} \\ \alpha^{80} & \alpha^{210} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3A | $3$ | $2$ | $C_3^5$ | 3A | 1A | $\left(\begin{array}{ll}0 & \alpha^{71} \\ \alpha^{50} & \alpha^{121} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3B | $3$ | $2$ | $C_3^5$ | 3B | 1A | $\left(\begin{array}{ll}\alpha & \alpha^{197} \\ \alpha^{176} & \alpha^{190} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3C | $3$ | $2$ | $C_3^5$ | 3C | 1A | $\left(\begin{array}{ll}\alpha^{2} & \alpha^{24} \\ \alpha^{3} & \alpha^{167} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3D | $3$ | $2$ | $C_3^5$ | 3D | 1A | $\left(\begin{array}{ll}\alpha^{3} & \alpha^{207} \\ \alpha^{186} & \alpha^{86} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3E | $3$ | $2$ | $C_3^5$ | 3E | 1A | $\left(\begin{array}{ll}\alpha^{4} & \alpha^{70} \\ \alpha^{49} & \alpha^{68} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3F | $3$ | $2$ | $C_3^5$ | 3F | 1A | $\left(\begin{array}{ll}\alpha^{5} & \alpha^{19} \\ \alpha^{240} & \alpha^{122} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3G | $3$ | $2$ | $C_3^5$ | 3G | 1A | $\left(\begin{array}{ll}\alpha^{6} & \alpha^{172} \\ \alpha^{151} & \alpha^{17} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3H | $3$ | $2$ | $C_3^5$ | 3H | 1A | $\left(\begin{array}{ll}\alpha^{7} & \alpha^{228} \\ \alpha^{207} & \alpha^{233} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3I | $3$ | $2$ | $C_3^5$ | 3I | 1A | $\left(\begin{array}{ll}\alpha^{8} & \alpha^{17} \\ \alpha^{238} & \alpha^{220} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3J | $3$ | $2$ | $C_3^5$ | 3J | 1A | $\left(\begin{array}{ll}\alpha^{9} & \alpha^{237} \\ \alpha^{216} & \alpha^{16} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3K | $3$ | $2$ | $C_3^5$ | 3K | 1A | $\left(\begin{array}{ll}\alpha^{10} & \alpha^{20} \\ \alpha^{241} & \alpha^{88} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3L | $3$ | $2$ | $C_3^5$ | 3L | 1A | $\left(\begin{array}{ll}\alpha^{11} & \alpha^{166} \\ \alpha^{145} & \alpha^{236} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3M | $3$ | $2$ | $C_3^5$ | 3M | 1A | $\left(\begin{array}{ll}\alpha^{12} & \alpha^{68} \\ \alpha^{47} & \alpha^{204} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3N | $3$ | $2$ | $C_3^5$ | 3N | 1A | $\left(\begin{array}{ll}\alpha^{13} & \alpha^{226} \\ \alpha^{205} & \alpha^{107} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3O | $3$ | $2$ | $C_3^5$ | 3O | 1A | $\left(\begin{array}{ll}\alpha^{14} & \alpha^{98} \\ \alpha^{77} & \alpha^{169} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3P | $3$ | $2$ | $C_3^5$ | 3P | 1A | $\left(\begin{array}{ll}\alpha^{15} & \alpha^{157} \\ \alpha^{136} & \alpha^{124} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3Q | $3$ | $2$ | $C_3^5$ | 3Q | 1A | $\left(\begin{array}{ll}\alpha^{18} & \alpha^{132} \\ \alpha^{111} & \alpha^{51} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3R | $3$ | $2$ | $C_3^5$ | 3R | 1A | $\left(\begin{array}{ll}\alpha^{19} & \alpha^{8} \\ \alpha^{229} & \alpha^{219} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3S | $3$ | $2$ | $C_3^5$ | 3S | 1A | $\left(\begin{array}{ll}\alpha^{20} & \alpha^{229} \\ \alpha^{208} & \alpha^{147} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3T | $3$ | $2$ | $C_3^5$ | 3T | 1A | $\left(\begin{array}{ll}\alpha^{21} & \alpha^{58} \\ \alpha^{37} & \alpha^{215} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3U | $3$ | $2$ | $C_3^5$ | 3U | 1A | $\left(\begin{array}{ll}\alpha^{22} & \alpha^{39} \\ \alpha^{18} & \alpha^{30} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3V | $3$ | $2$ | $C_3^5$ | 3V | 1A | $\left(\begin{array}{ll}\alpha^{23} & \alpha^{31} \\ \alpha^{10} & \alpha^{42} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3W | $3$ | $2$ | $C_3^5$ | 3W | 1A | $\left(\begin{array}{ll}\alpha^{24} & \alpha^{151} \\ \alpha^{130} & \alpha^{176} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3X | $3$ | $2$ | $C_3^5$ | 3X | 1A | $\left(\begin{array}{ll}\alpha^{25} & \alpha^{161} \\ \alpha^{140} & \alpha^{187} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3Y | $3$ | $2$ | $C_3^5$ | 3Y | 1A | $\left(\begin{array}{ll}\alpha^{26} & \alpha^{212} \\ \alpha^{191} & \alpha^{158} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3Z | $3$ | $2$ | $C_3^5$ | 3Z | 1A | $\left(\begin{array}{ll}\alpha^{27} & \alpha^{85} \\ \alpha^{64} & \alpha^{48} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AA | $3$ | $2$ | $C_3^5$ | 3AA | 1A | $\left(\begin{array}{ll}\alpha^{28} & \alpha^{146} \\ \alpha^{125} & \alpha^{198} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AB | $3$ | $2$ | $C_3^5$ | 3AB | 1A | $\left(\begin{array}{ll}\alpha^{29} & \alpha^{64} \\ \alpha^{43} & \alpha^{105} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AC | $3$ | $2$ | $C_3^5$ | 3AC | 1A | $\left(\begin{array}{ll}\alpha^{31} & \alpha^{168} \\ \alpha^{147} & \alpha^{177} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AD | $3$ | $2$ | $C_3^5$ | 3AD | 1A | $\left(\begin{array}{ll}\alpha^{32} & \alpha^{4} \\ \alpha^{225} & \alpha^{183} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AE | $3$ | $2$ | $C_3^5$ | 3AE | 1A | $\left(\begin{array}{ll}\alpha^{33} & \alpha^{114} \\ \alpha^{93} & \alpha^{224} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AF | $3$ | $2$ | $C_3^5$ | 3AF | 1A | $\left(\begin{array}{ll}\alpha^{34} & \alpha^{178} \\ \alpha^{157} & \alpha^{134} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AG | $3$ | $2$ | $C_3^5$ | 3AG | 1A | $\left(\begin{array}{ll}\alpha^{35} & \alpha^{109} \\ \alpha^{88} & \alpha^{171} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AH | $3$ | $2$ | $C_3^5$ | 3AH | 1A | $\left(\begin{array}{ll}\alpha^{36} & \alpha^{62} \\ \alpha^{41} & \alpha^{128} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AI | $3$ | $2$ | $C_3^5$ | 3AI | 1A | $\left(\begin{array}{ll}\alpha^{37} & \alpha^{218} \\ \alpha^{197} & \alpha^{141} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AJ | $3$ | $2$ | $C_3^5$ | 3AJ | 1A | $\left(\begin{array}{ll}\alpha^{38} & \alpha^{106} \\ \alpha^{85} & \alpha^{50} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AK | $3$ | $2$ | $C_3^5$ | 3AK | 1A | $\left(\begin{array}{ll}\alpha^{39} & \alpha^{52} \\ \alpha^{31} & \alpha^{79} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AL | $3$ | $2$ | $C_3^5$ | 3AL | 1A | $\left(\begin{array}{ll}\alpha^{40} & \alpha^{13} \\ \alpha^{234} & \alpha^{203} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AM | $3$ | $2$ | $C_3^5$ | 3AM | 1A | $\left(\begin{array}{ll}\alpha^{41} & \alpha^{167} \\ \alpha^{146} & \alpha^{186} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AN | $3$ | $2$ | $C_3^5$ | 3AN | 1A | $\left(\begin{array}{ll}\alpha^{43} & \alpha^{104} \\ \alpha^{83} & \alpha^{103} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AO | $3$ | $2$ | $C_3^5$ | 3AO | 1A | $\left(\begin{array}{ll}\alpha^{44} & \alpha^{190} \\ \alpha^{169} & \alpha^{72} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AP | $3$ | $2$ | $C_3^5$ | 3AP | 1A | $\left(\begin{array}{ll}\alpha^{45} & \alpha^{87} \\ \alpha^{66} & \alpha^{130} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AQ | $3$ | $2$ | $C_3^5$ | 3AQ | 1A | $\left(\begin{array}{ll}\alpha^{46} & \alpha^{194} \\ \alpha^{173} & \alpha^{195} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AR | $3$ | $2$ | $C_3^5$ | 3AR | 1A | $\left(\begin{array}{ll}\alpha^{47} & \alpha^{241} \\ \alpha^{220} & \alpha^{93} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AS | $3$ | $2$ | $C_3^5$ | 3AS | 1A | $\left(\begin{array}{ll}\alpha^{49} & \alpha^{164} \\ \alpha^{143} & \alpha^{168} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AT | $3$ | $2$ | $C_3^5$ | 3AT | 1A | $\left(\begin{array}{ll}\alpha^{52} & \alpha^{7} \\ \alpha^{228} & \alpha^{53} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AU | $3$ | $2$ | $C_3^5$ | 3AU | 1A | $\left(\begin{array}{ll}\alpha^{54} & \alpha^{12} \\ \alpha^{233} & \alpha^{153} \\ \end{array}\right)$ |
| $C_3^4:S_3$ | 3AV | $3$ | $2$ | $C_3^5$ | 3AV | 1A | $\left(\begin{array}{ll}\alpha^{55} & \alpha^{216} \\ \alpha^{195} & \alpha^{80} \\ \end{array}\right)$ |