Abstract group 531360.a: $\SL(2,81)$

• Label: 531360.a
• Order: \( 2^{5} \cdot 3^{4} \cdot 5 \cdot 41 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 5 \cdot 41 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{4} \cdot 5 \cdot 41 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $1312$
• Trans deg.: not computed
• Rank: $2$

Group information

Description:	$\SL(2,81)$
Order:	\(531360\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \cdot 41 \)
Exponent:	\(9840\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 41 \)
Automorphism group:	$\PGammaL(2,81)$, of order \(2125440\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \cdot 41 \)
Composition factors:	$C_2$, $\PSL(2,81)$
Derived length:	$0$

This group is nonabelian and quasisimple (hence nonsolvable and perfect). Whether it is almost simple has not been computed.

Group statistics

Order	1	2	3	4	5	6	8	10	16	20	40	41	80	82
Elements	1	1	6560	6642	13284	6560	13284	13284	26568	26568	53136	129600	106272	129600	531360
Conjugacy classes	1	1	2	1	2	2	2	2	4	4	8	20	16	20	85
Divisions	1	1	2	1	1	2	1	1	1	1	1	1	1	1	16
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	2	5	4	5	26

Minimal presentations

Permutation degree:	$1312$
Transitive degree:	not computed
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	40	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\SL(2,81)$, $\SU(2,81)$, $\SpinMinus(4,9)$
Permutation group:	Degree $1312$ $\langle(1,3,5)(2,7,8)(4,11,13)(6,15,16)(9,20,21)(10,22,24)(12,27,29)(14,31,32) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{ll}\alpha^{47} & \alpha^{42} \\ \alpha^{8} & \alpha^{63} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{13} & \alpha^{41} \\ 0 & \alpha^{67} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{40} & 0 \\ 0 & \alpha^{40} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{7} & \alpha^{78} \\ \alpha^{44} & \alpha^{58} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{46} & \alpha^{62} \\ \alpha^{35} & \alpha^{49} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{69} & \alpha^{34} \\ \alpha^{48} & \alpha^{69} \\ \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{81}) = \GL_{2}(\F_{3}[\alpha]/(\alpha^{4} + 2 \alpha^{3} + 2))$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\SL(2,81)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

Subgroup data has not been computed.

Character theory

Complex character table

The $85 \times 85$ character table is not available for this group.

Rational character table

The $16 \times 16$ rational character table is not available for this group.