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Elements of the group are displayed as matrices in $\GL_{2}(\F_{81})$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | ||
|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 5P | ||||||
| $F_{81}$ | 1A | $1$ | $1$ | $F_{81}$ | 1A | 1A | 1A | $\left(\begin{array}{ll}1 & 0 \\ 0 & 1 \\ \end{array}\right)$ |
| $F_{81}$ | 2A | $2$ | $81$ | $C_{80}$ | 1A | 2A | 2A | $\left(\begin{array}{ll}\alpha^{47} & \alpha^{63} \\ \alpha^{63} & \alpha^{7} \\ \end{array}\right)$ |
| $F_{81}$ | 3A | $3$ | $80$ | $C_3^4$ | 3A | 1A | 3A | $\left(\begin{array}{ll}0 & \alpha \\ \alpha^{39} & \alpha^{40} \\ \end{array}\right)$ |
| $F_{81}$ | 4A1 | $4$ | $81$ | $C_{80}$ | 2A | 4A-1 | 4A1 | $\left(\begin{array}{ll}\alpha^{66} & \alpha^{73} \\ \alpha^{73} & \alpha^{69} \\ \end{array}\right)$ |
| $F_{81}$ | 4A-1 | $4$ | $81$ | $C_{80}$ | 2A | 4A1 | 4A-1 | $\left(\begin{array}{ll}\alpha^{49} & \alpha^{13} \\ \alpha^{13} & \alpha^{46} \\ \end{array}\right)$ |
| $F_{81}$ | 5A1 | $5$ | $81$ | $C_{80}$ | 5A2 | 5A-2 | 1A | $\left(\begin{array}{ll}\alpha^{20} & \alpha^{60} \\ \alpha^{60} & \alpha^{27} \\ \end{array}\right)$ |
| $F_{81}$ | 5A-1 | $5$ | $81$ | $C_{80}$ | 5A-2 | 5A2 | 1A | $\left(\begin{array}{ll}\alpha^{43} & \alpha^{36} \\ \alpha^{36} & \alpha^{36} \\ \end{array}\right)$ |
| $F_{81}$ | 5A2 | $5$ | $81$ | $C_{80}$ | 5A-1 | 5A1 | 1A | $\left(\begin{array}{ll}1 & \alpha^{62} \\ \alpha^{62} & \alpha^{35} \\ \end{array}\right)$ |
| $F_{81}$ | 5A-2 | $5$ | $81$ | $C_{80}$ | 5A1 | 5A-1 | 1A | $\left(\begin{array}{ll}\alpha^{67} & \alpha^{54} \\ \alpha^{54} & \alpha^{32} \\ \end{array}\right)$ |
| $F_{81}$ | 8A1 | $8$ | $81$ | $C_{80}$ | 4A1 | 8A3 | 8A-3 | $\left(\begin{array}{ll}\alpha^{53} & \alpha^{53} \\ \alpha^{53} & \alpha^{75} \\ \end{array}\right)$ |
| $F_{81}$ | 8A-1 | $8$ | $81$ | $C_{80}$ | 4A-1 | 8A-3 | 8A3 | $\left(\begin{array}{ll}\alpha^{65} & \alpha^{3} \\ \alpha^{3} & \alpha^{43} \\ \end{array}\right)$ |
| $F_{81}$ | 8A3 | $8$ | $81$ | $C_{80}$ | 4A-1 | 8A1 | 8A-1 | $\left(\begin{array}{ll}\alpha^{21} & \alpha^{33} \\ \alpha^{33} & \alpha^{8} \\ \end{array}\right)$ |
| $F_{81}$ | 8A-3 | $8$ | $81$ | $C_{80}$ | 4A1 | 8A-1 | 8A1 | $\left(\begin{array}{ll}\alpha^{58} & \alpha^{43} \\ \alpha^{43} & \alpha^{71} \\ \end{array}\right)$ |
| $F_{81}$ | 10A1 | $10$ | $81$ | $C_{80}$ | 5A1 | 10A3 | 2A | $\left(\begin{array}{ll}\alpha^{38} & \alpha^{29} \\ \alpha^{29} & \alpha^{15} \\ \end{array}\right)$ |
| $F_{81}$ | 10A-1 | $10$ | $81$ | $C_{80}$ | 5A-1 | 10A-3 | 2A | $\left(\begin{array}{ll}\alpha^{23} & \alpha^{77} \\ \alpha^{77} & \alpha^{46} \\ \end{array}\right)$ |
| $F_{81}$ | 10A3 | $10$ | $81$ | $C_{80}$ | 5A-2 | 10A-1 | 2A | $\left(\begin{array}{ll}\alpha^{22} & \alpha^{41} \\ \alpha^{41} & \alpha^{26} \\ \end{array}\right)$ |
| $F_{81}$ | 10A-3 | $10$ | $81$ | $C_{80}$ | 5A2 | 10A1 | 2A | $\left(\begin{array}{ll}\alpha^{50} & \alpha^{25} \\ \alpha^{25} & \alpha^{46} \\ \end{array}\right)$ |
| $F_{81}$ | 16A1 | $16$ | $81$ | $C_{80}$ | 8A1 | 16A3 | 16A5 | $\left(\begin{array}{ll}\alpha^{55} & \alpha^{34} \\ \alpha^{34} & \alpha^{6} \\ \end{array}\right)$ |
| $F_{81}$ | 16A-1 | $16$ | $81$ | $C_{80}$ | 8A-1 | 16A-3 | 16A-5 | $\left(\begin{array}{ll}\alpha^{41} & \alpha^{29} \\ \alpha^{29} & \alpha^{10} \\ \end{array}\right)$ |
| $F_{81}$ | 16A3 | $16$ | $81$ | $C_{80}$ | 8A3 | 16A-7 | 16A-1 | $\left(\begin{array}{ll}\alpha^{33} & \alpha^{56} \\ \alpha^{56} & \alpha^{15} \\ \end{array}\right)$ |
| $F_{81}$ | 16A-3 | $16$ | $81$ | $C_{80}$ | 8A-3 | 16A7 | 16A1 | $\left(\begin{array}{ll}\alpha^{40} & \alpha^{41} \\ \alpha^{41} & \alpha^{58} \\ \end{array}\right)$ |
| $F_{81}$ | 16A5 | $16$ | $81$ | $C_{80}$ | 8A-3 | 16A-1 | 16A-7 | $\left(\begin{array}{ll}\alpha^{22} & \alpha^{25} \\ \alpha^{25} & \alpha^{61} \\ \end{array}\right)$ |
| $F_{81}$ | 16A-5 | $16$ | $81$ | $C_{80}$ | 8A3 | 16A1 | 16A7 | $\left(\begin{array}{ll}\alpha^{76} & 1 \\ 1 & \alpha^{37} \\ \end{array}\right)$ |
| $F_{81}$ | 16A7 | $16$ | $81$ | $C_{80}$ | 8A-1 | 16A5 | 16A3 | $\left(\begin{array}{ll}\alpha^{79} & \alpha^{77} \\ \alpha^{77} & \alpha^{23} \\ \end{array}\right)$ |
| $F_{81}$ | 16A-7 | $16$ | $81$ | $C_{80}$ | 8A1 | 16A-5 | 16A-3 | $\left(\begin{array}{ll}\alpha^{28} & \alpha^{42} \\ \alpha^{42} & \alpha^{4} \\ \end{array}\right)$ |
| $F_{81}$ | 20A1 | $20$ | $81$ | $C_{80}$ | 10A1 | 20A3 | 4A1 | $\left(\begin{array}{ll}\alpha^{69} & \alpha^{4} \\ \alpha^{4} & \alpha^{16} \\ \end{array}\right)$ |
| $F_{81}$ | 20A-1 | $20$ | $81$ | $C_{80}$ | 10A-1 | 20A-3 | 4A-1 | $\left(\begin{array}{ll}\alpha^{60} & \alpha^{8} \\ \alpha^{8} & \alpha^{33} \\ \end{array}\right)$ |
| $F_{81}$ | 20A3 | $20$ | $81$ | $C_{80}$ | 10A3 | 20A9 | 4A-1 | $\left(\begin{array}{ll}\alpha^{41} & \alpha^{46} \\ \alpha^{46} & \alpha^{9} \\ \end{array}\right)$ |
| $F_{81}$ | 20A-3 | $20$ | $81$ | $C_{80}$ | 10A-3 | 20A-9 | 4A1 | $\left(\begin{array}{ll}\alpha^{61} & \alpha^{58} \\ \alpha^{58} & \alpha^{13} \\ \end{array}\right)$ |
| $F_{81}$ | 20A7 | $20$ | $81$ | $C_{80}$ | 10A-3 | 20A1 | 4A-1 | $\left(\begin{array}{ll}1 & \alpha^{70} \\ \alpha^{70} & \alpha^{46} \\ \end{array}\right)$ |
| $F_{81}$ | 20A-7 | $20$ | $81$ | $C_{80}$ | 10A3 | 20A-1 | 4A1 | $\left(\begin{array}{ll}\alpha^{34} & \alpha^{18} \\ \alpha^{18} & \alpha^{68} \\ \end{array}\right)$ |
| $F_{81}$ | 20A9 | $20$ | $81$ | $C_{80}$ | 10A-1 | 20A7 | 4A1 | $\left(\begin{array}{ll}\alpha^{70} & \alpha^{12} \\ \alpha^{12} & \alpha^{4} \\ \end{array}\right)$ |
| $F_{81}$ | 20A-9 | $20$ | $81$ | $C_{80}$ | 10A1 | 20A-7 | 4A-1 | $\left(\begin{array}{ll}1 & \alpha^{48} \\ \alpha^{48} & \alpha^{66} \\ \end{array}\right)$ |
| $F_{81}$ | 40A1 | $40$ | $81$ | $C_{80}$ | 20A1 | 40A3 | 8A1 | $\left(\begin{array}{ll}\alpha^{42} & \alpha^{63} \\ \alpha^{63} & \alpha^{53} \\ \end{array}\right)$ |
| $F_{81}$ | 40A-1 | $40$ | $81$ | $C_{80}$ | 20A-1 | 40A-3 | 8A-1 | $\left(\begin{array}{ll}\alpha^{35} & \alpha^{5} \\ \alpha^{5} & \alpha^{24} \\ \end{array}\right)$ |
| $F_{81}$ | 40A3 | $40$ | $81$ | $C_{80}$ | 20A3 | 40A9 | 8A3 | $\left(\begin{array}{ll}\alpha^{30} & \alpha^{63} \\ \alpha^{63} & \alpha^{45} \\ \end{array}\right)$ |
| $F_{81}$ | 40A-3 | $40$ | $81$ | $C_{80}$ | 20A-3 | 40A-9 | 8A-3 | $\left(\begin{array}{ll}\alpha^{71} & \alpha^{49} \\ \alpha^{49} & \alpha^{56} \\ \end{array}\right)$ |
| $F_{81}$ | 40A7 | $40$ | $81$ | $C_{80}$ | 20A7 | 40A-19 | 8A-1 | $\left(\begin{array}{ll}\alpha^{75} & \alpha^{30} \\ \alpha^{30} & \alpha^{46} \\ \end{array}\right)$ |
| $F_{81}$ | 40A-7 | $40$ | $81$ | $C_{80}$ | 20A-7 | 40A19 | 8A1 | $\left(\begin{array}{ll}1 & \alpha^{24} \\ \alpha^{24} & \alpha^{29} \\ \end{array}\right)$ |
| $F_{81}$ | 40A9 | $40$ | $81$ | $C_{80}$ | 20A9 | 40A-13 | 8A1 | $\left(\begin{array}{ll}\alpha^{14} & \alpha^{63} \\ \alpha^{63} & \alpha^{31} \\ \end{array}\right)$ |
| $F_{81}$ | 40A-9 | $40$ | $81$ | $C_{80}$ | 20A-9 | 40A13 | 8A-1 | $\left(\begin{array}{ll}\alpha^{29} & \alpha^{21} \\ \alpha^{21} & \alpha^{12} \\ \end{array}\right)$ |
| $F_{81}$ | 40A11 | $40$ | $81$ | $C_{80}$ | 20A-9 | 40A-7 | 8A3 | $\left(\begin{array}{ll}\alpha^{25} & \alpha^{50} \\ \alpha^{50} & \alpha^{37} \\ \end{array}\right)$ |
| $F_{81}$ | 40A-11 | $40$ | $81$ | $C_{80}$ | 20A9 | 40A7 | 8A-3 | $\left(\begin{array}{ll}\alpha^{79} & \alpha^{52} \\ \alpha^{52} & \alpha^{67} \\ \end{array}\right)$ |
| $F_{81}$ | 40A13 | $40$ | $81$ | $C_{80}$ | 20A-7 | 40A-1 | 8A-3 | $\left(\begin{array}{ll}0 & \alpha^{17} \\ \alpha^{17} & \alpha \\ \end{array}\right)$ |
| $F_{81}$ | 40A-13 | $40$ | $81$ | $C_{80}$ | 20A7 | 40A1 | 8A3 | $\left(\begin{array}{ll}\alpha^{7} & \alpha^{63} \\ \alpha^{63} & 0 \\ \end{array}\right)$ |
| $F_{81}$ | 40A17 | $40$ | $81$ | $C_{80}$ | 20A-3 | 40A11 | 8A1 | $\left(\begin{array}{ll}\alpha^{46} & \alpha^{32} \\ \alpha^{32} & \alpha^{76} \\ \end{array}\right)$ |
| $F_{81}$ | 40A-17 | $40$ | $81$ | $C_{80}$ | 20A3 | 40A-11 | 8A-1 | $\left(\begin{array}{ll}\alpha^{10} & \alpha^{6} \\ \alpha^{6} & \alpha^{60} \\ \end{array}\right)$ |
| $F_{81}$ | 40A19 | $40$ | $81$ | $C_{80}$ | 20A-1 | 40A17 | 8A3 | $\left(\begin{array}{ll}\alpha^{63} & \alpha^{26} \\ \alpha^{26} & \alpha^{19} \\ \end{array}\right)$ |
| $F_{81}$ | 40A-19 | $40$ | $81$ | $C_{80}$ | 20A1 | 40A-17 | 8A-3 | $\left(\begin{array}{ll}\alpha^{77} & \alpha^{44} \\ \alpha^{44} & \alpha^{41} \\ \end{array}\right)$ |
| $F_{81}$ | 80A1 | $80$ | $81$ | $C_{80}$ | 40A1 | 80A3 | 16A1 | $\left(\begin{array}{ll}\alpha^{17} & \alpha^{54} \\ \alpha^{54} & \alpha^{43} \\ \end{array}\right)$ |