Abstract group 1285608.a: $\PSL(2,137)$

• Label: 1285608.a
• Order: \( 2^{3} \cdot 3 \cdot 17 \cdot 23 \cdot 137 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 17 \cdot 23 \cdot 137 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 17 \cdot 23 \cdot 137 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $138$
• Trans deg.: $138$
• Rank: $2$

Group information

Description:	$\PSL(2,137)$
Order:	\(1285608\)\(\medspace = 2^{3} \cdot 3 \cdot 17 \cdot 23 \cdot 137 \)
Exponent:	\(642804\)\(\medspace = 2^{2} \cdot 3 \cdot 17 \cdot 23 \cdot 137 \)
Automorphism group:	$\PGL(2,137)$, of order \(2571216\)\(\medspace = 2^{4} \cdot 3 \cdot 17 \cdot 23 \cdot 137 \)
Composition factors:	$\PSL(2,137)$
Derived length:	$0$

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

Group statistics

Order	1	2	3	4	17	23	34	68	69	137
Elements	1	9453	18632	18906	151248	204952	151248	302496	409904	18768	1285608
Conjugacy classes	1	1	1	1	8	11	8	16	22	2	71
Divisions	1	1	1	1	1	1	1	1	1	1	10
Autjugacy classes	1	1	1	1	8	11	8	16	22	1	70

Dimension	1	69	136	137	138	1104	1496	2208	2992
Irr. complex chars.	1	2	34	1	33	0	0	0	0	71
Irr. rational chars.	1	0	1	1	2	2	1	1	1	10

Minimal presentations

Permutation degree:	$138$
Transitive degree:	$138$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	69	69	136
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PSL(2,137)$, $\PSU(2,137)$, $\Omega(3,137)$, $\POmega(3,137)$, $\PSigmaL(2,137)$
Permutation group:	Degree $138$ $\langle(1,71,2)(3,86,124)(4,7,10)(5,132,24)(6,41,116)(8,36,111)(9,128,20)(11,28,66) \!\cdots\! \rangle$
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 $\SL(2,137)$.

Homology

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

Subgroups

There are 875849 subgroups in 31 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\PSL(2,137)$
Commutator:	$G' \simeq$ $\PSL(2,137)$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\PSL(2,137)$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\PSL(2,137)$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\PSL(2,137)$
Socle:	$\operatorname{soc} \simeq$ $\PSL(2,137)$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$
23-Sylow subgroup:	$P_{ 23 } \simeq$ $C_{23}$
137-Sylow subgroup:	$P_{ 137 } \simeq$ $C_{137}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 1285608: $\PSL(2,137)$

Order 9316: $C_{137}:C_{68}$

Order 4658: $C_{137}:C_{34}$

Order 2329: $C_{137}:C_{17}$

Order 548: $C_{137}:C_4$

Order 274: $D_{137}$

Order 138: $D_{69}$

Order 137: $C_{137}$

Order 136: $D_{68}$

Order 69: $C_{69}$

Order 68: $D_{34}$ x 2, $C_{68}$

Order 46: $D_{23}$

Order 34: $D_{17}$ x 2, $C_{34}$

Order 24: $S_4$ x 2

Order 23: $C_{23}$

Order 17: $C_{17}$

Order 12: $A_4$ x 2

Order 8: $D_4$

Order 6: $S_3$

Order 4: $C_2^2$ x 2, $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1285608: $\PSL(2,137)$

Order 9316: $C_{137}:C_{68}$

Order 4658: $C_{137}:C_{34}$

Order 2329: $C_{137}:C_{17}$

Order 548: $C_{137}:C_4$

Order 274: $D_{137}$

Order 138: $D_{69}$

Order 137: $C_{137}$

Order 136: $D_{68}$

Order 69: $C_{69}$

Order 68: $C_{68}$

Order 46: $D_{23}$

Order 23: $C_{23}$

Order 17: $C_{17}$

Order 8: $D_4$

Order 6: $S_3$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1285608: $\PSL(2,137)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,137)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1285608: $\PSL(2,137)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,137)$)

Series

Derived series	$\PSL(2,137)$
Chief series	$\PSL(2,137)$	$\rhd$	$C_1$
Lower central series	$\PSL(2,137)$
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 $71 \times 71$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	4A	17A	23A	34A	68A	69A	137A
	Size	1	9453	18632	18906	151248	204952	151248	302496	409904	18768
	2 P	1A	1A	3A	2A	17A	23A	17A	34A	69A	137A
	3 P	1A	2A	1A	4A	17A	23A	34A	68A	23A	137A
	17 P	1A	2A	3A	4A	1A	23A	2A	4A	69A	137A
	23 P	1A	2A	3A	4A	17A	1A	34A	68A	3A	137A
	137 P	1A	2A	3A	4A	17A	23A	34A	68A	69A	1A
1285608.a.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1285608.a.69a		$138$	$2$	$0$	$-2$	$2$	$0$	$2$	$-2$	$0$	$1$
1285608.a.136a		$136$	$0$	$1$	$0$	$0$	$-2$	$0$	$0$	$1$	$-1$
1285608.a.136b		$1496$	$0$	$-22$	$0$	$0$	$1$	$0$	$0$	$1$	$-11$
1285608.a.136c		$2992$	$0$	$22$	$0$	$0$	$2$	$0$	$0$	$-1$	$-22$
1285608.a.137a		$137$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$0$
1285608.a.138a		$138$	$-2$	$0$	$0$	$2$	$0$	$-2$	$0$	$0$	$1$
1285608.a.138b		$1104$	$16$	$0$	$16$	$-1$	$0$	$-1$	$-1$	$0$	$8$
1285608.a.138c		$1104$	$16$	$0$	$-16$	$-1$	$0$	$-1$	$1$	$0$	$8$
1285608.a.138d		$2208$	$-32$	$0$	$0$	$-2$	$0$	$2$	$0$	$0$	$16$