Abstract group 1123980.a: $\PSL(2,131)$

• Label: 1123980.a
• Order: \( 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \cdot 131 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 11 \cdot 13 \cdot 131 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \cdot 131 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $132$
• Trans deg.: $132$
• Rank: $2$

Group information

Description:	$\PSL(2,131)$
Order:	\(1123980\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \cdot 131 \)
Exponent:	\(561990\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 13 \cdot 131 \)
Automorphism group:	$\PGL(2,131)$, of order \(2247960\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \cdot 131 \)
Composition factors:	$\PSL(2,131)$
Derived length:	$0$

This group is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.

Group statistics

Order	1	2	3	5	6	11	13	22	33	65	66	131
Elements	1	8515	17030	34584	17030	85150	103752	85150	170300	415008	170300	17160	1123980
Conjugacy classes	1	1	1	2	1	5	6	5	10	24	10	2	68
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	12
Autjugacy classes	1	1	1	2	1	5	6	5	10	24	10	1	67

Dimension	1	65	130	131	132	264	650	792	1300	3168
Irr. complex chars.	1	2	32	1	32	0	0	0	0	0	68
Irr. rational chars.	1	0	3	1	0	1	2	1	2	1	12

Minimal presentations

Permutation degree:	$132$
Transitive degree:	$132$
Rank:	$2$
Inequivalent generating pairs:	$557594$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	65	130	130
Arbitrary	65	130	130

Constructions

Groups of Lie type:	$\PSL(2,131)$, $\PSU(2,131)$, $\Omega(3,131)$, $\POmega(3,131)$, $\PSigmaL(2,131)$
Permutation group:	Degree $132$ $\langle(1,68,2)(3,109,67)(4,17,47)(5,23,80)(6,117,40)(7,60,79)(8,19,128)(9,20,32) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\SL(2,131)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 858318 subgroups in 32 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\PSL(2,131)$
Commutator:	$G' \simeq$ $\PSL(2,131)$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\PSL(2,131)$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\PSL(2,131)$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\PSL(2,131)$
Socle:	$\operatorname{soc} \simeq$ $\PSL(2,131)$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$
131-Sylow subgroup:	$P_{ 131 } \simeq$ $C_{131}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1123980: $\PSL(2,131)$

Order 8515: $C_{131}:C_{65}$

Order 1703: $C_{131}:C_{13}$

Order 655: $C_{131}:C_5$

Order 132: $D_{66}$

Order 131: $C_{131}$

Order 130: $D_{65}$

Order 66: $D_{33}$ x 2, $C_{66}$

Order 65: $C_{65}$

Order 60: $A_5$ x 2

Order 44: $D_{22}$

Order 33: $C_{33}$

Order 26: $D_{13}$

Order 22: $D_{11}$ x 2, $C_{22}$

Order 13: $C_{13}$

Order 12: $D_6$, $A_4$

Order 11: $C_{11}$

Order 10: $D_5$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1123980: $\PSL(2,131)$

Order 8515: $C_{131}:C_{65}$

Order 1703: $C_{131}:C_{13}$

Order 655: $C_{131}:C_5$

Order 132: $D_{66}$

Order 131: $C_{131}$

Order 130: $D_{65}$

Order 66: $C_{66}$, $D_{33}$

Order 65: $C_{65}$

Order 60: $A_5$

Order 44: $D_{22}$

Order 33: $C_{33}$

Order 26: $D_{13}$

Order 22: $C_{22}$, $D_{11}$

Order 13: $C_{13}$

Order 12: $D_6$, $A_4$

Order 11: $C_{11}$

Order 10: $D_5$

Order 6: $C_6$, $S_3$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1123980: $\PSL(2,131)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,131)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1123980: $\PSL(2,131)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,131)$)

Series

Derived series	$\PSL(2,131)$
Chief series	$\PSL(2,131)$	$\rhd$	$C_1$
Lower central series	$\PSL(2,131)$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $68 \times 68$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	5A	6A	11A	13A	22A	33A	65A	66A	131A
	Size	1	8515	17030	34584	17030	85150	103752	85150	170300	415008	170300	17160
	2 P	1A	1A	3A	5A	3A	11A	13A	11A	33A	65A	33A	131A
	3 P	1A	2A	1A	5A	2A	11A	13A	22A	11A	65A	22A	131A
	5 P	1A	2A	3A	1A	6A	11A	13A	22A	33A	13A	66A	131A
	11 P	1A	2A	3A	5A	6A	1A	13A	2A	3A	65A	6A	131A
	13 P	1A	2A	3A	5A	6A	11A	1A	22A	33A	5A	66A	131A
	131 P	1A	2A	3A	5A	6A	11A	13A	22A	33A	65A	66A	1A
1123980.a.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1123980.a.65a		$130$	$2$	$-2$	$0$	$2$	$-2$	$0$	$2$	$-2$	$0$	$2$	$-1$
1123980.a.130a		$130$	$2$	$1$	$0$	$-1$	$-2$	$0$	$2$	$1$	$0$	$-1$	$-1$
1123980.a.130b		$130$	$-2$	$1$	$0$	$1$	$-2$	$0$	$-2$	$1$	$0$	$1$	$-1$
1123980.a.130c		$650$	$10$	$-10$	$0$	$10$	$1$	$0$	$-1$	$1$	$0$	$-1$	$-5$
1123980.a.130d		$650$	$-10$	$-10$	$0$	$-10$	$1$	$0$	$1$	$1$	$0$	$1$	$-5$
1123980.a.130e		$1300$	$20$	$10$	$0$	$-10$	$2$	$0$	$-2$	$-1$	$0$	$1$	$-10$
1123980.a.130f		$1300$	$-20$	$10$	$0$	$10$	$2$	$0$	$2$	$-1$	$0$	$-1$	$-10$
1123980.a.131a		$131$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$0$
1123980.a.132a		$264$	$0$	$0$	$-1$	$0$	$0$	$4$	$0$	$0$	$-1$	$0$	$2$
1123980.a.132b		$792$	$0$	$0$	$12$	$0$	$0$	$-1$	$0$	$0$	$-1$	$0$	$6$
1123980.a.132c		$3168$	$0$	$0$	$-12$	$0$	$0$	$-4$	$0$	$0$	$1$	$0$	$24$