Abstract group 102660.a: $\PSL(2,59)$

• Label: 102660.a
• Order: \( 2^{2} \cdot 3 \cdot 5 \cdot 29 \cdot 59 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 29 \cdot 59 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3 \cdot 5 \cdot 29 \cdot 59 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $60$
• Trans deg.: $60$
• Rank: $2$

Group information

Description:	$\PSL(2,59)$
Order:	\(102660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 29 \cdot 59 \)
Exponent:	\(51330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 29 \cdot 59 \)
Automorphism group:	$\PGL(2,59)$, of order \(205320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 29 \cdot 59 \)
Composition factors:	$\PSL(2,59)$
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	10	15	29	30	59
Elements	1	1711	3422	6844	3422	6844	13688	49560	13688	3480	102660
Conjugacy classes	1	1	1	2	1	2	4	14	4	2	32
Divisions	1	1	1	1	1	1	1	1	1	1	10
Autjugacy classes	1	1	1	2	1	2	4	14	4	1	31

Dimension	1	29	58	59	60	116	232	840
Irr. complex chars.	1	2	14	1	14	0	0	0	32
Irr. rational chars.	1	0	3	1	0	2	2	1	10

Minimal presentations

Permutation degree:	$60$
Transitive degree:	$60$
Rank:	$2$
Inequivalent generating pairs:	$50390$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	29	58	58
Arbitrary	29	58	58

Constructions

Groups of Lie type:	$\PSL(2,59)$, $\PSU(2,59)$, $\Omega(3,59)$, $\POmega(3,59)$, $\PSigmaL(2,59)$
Permutation group:	Degree $60$ $\langle(1,60,2)(3,31,59)(4,20,57)(5,42,58)(6,46,28)(7,26,17)(8,52,39)(9,30,43) \!\cdots\! \rangle$
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 $\SL(2,59)$.

Homology

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

Subgroups

There are 82368 subgroups in 26 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\PSL(2,59)$
Commutator:	$G' \simeq$ $\PSL(2,59)$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\PSL(2,59)$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\PSL(2,59)$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\PSL(2,59)$
Socle:	$\operatorname{soc} \simeq$ $\PSL(2,59)$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
29-Sylow subgroup:	$P_{ 29 } \simeq$ $C_{29}$
59-Sylow subgroup:	$P_{ 59 } \simeq$ $C_{59}$

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 102660: $\PSL(2,59)$

Order 1711: $C_{59}:C_{29}$

Order 60: $A_5$ x 2, $D_{30}$

Order 59: $C_{59}$

Order 58: $D_{29}$

Order 30: $D_{15}$ x 2, $C_{30}$

Order 29: $C_{29}$

Order 20: $D_{10}$

Order 15: $C_{15}$

Order 12: $D_6$, $A_4$

Order 10: $D_5$ x 2, $C_{10}$

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 102660: $\PSL(2,59)$

Order 1711: $C_{59}:C_{29}$

Order 60: $A_5$, $D_{30}$

Order 59: $C_{59}$

Order 58: $D_{29}$

Order 30: $C_{30}$, $D_{15}$

Order 29: $C_{29}$

Order 20: $D_{10}$

Order 15: $C_{15}$

Order 12: $D_6$, $A_4$

Order 10: $C_{10}$, $D_5$

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 102660: $\PSL(2,59)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,59)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 102660: $\PSL(2,59)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,59)$)

Series

Derived series	$\PSL(2,59)$
Chief series	$\PSL(2,59)$	$\rhd$	$C_1$
Lower central series	$\PSL(2,59)$
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 $32 \times 32$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	5A	6A	10A	15A	29A	30A	59A
	Size	1	1711	3422	6844	3422	6844	13688	49560	13688	3480
	2 P	1A	1A	3A	5A	3A	5A	15A	29A	15A	59A
	3 P	1A	2A	1A	5A	2A	10A	5A	29A	10A	59A
	5 P	1A	2A	3A	1A	6A	2A	3A	29A	6A	59A
	29 P	1A	2A	3A	5A	6A	10A	15A	29A	30A	59A
	59 P	1A	2A	3A	5A	6A	10A	15A	1A	30A	59A
102660.a.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
102660.a.29a		$58$	$2$	$-2$	$-2$	$2$	$2$	$-2$	$0$	$2$	$-1$
102660.a.58a		$58$	$2$	$1$	$-2$	$-1$	$2$	$1$	$0$	$-1$	$-1$
102660.a.58b		$58$	$-2$	$1$	$-2$	$1$	$-2$	$1$	$0$	$1$	$-1$
102660.a.58c		$116$	$4$	$-4$	$1$	$4$	$-1$	$1$	$0$	$-1$	$-2$
102660.a.58d		$116$	$-4$	$-4$	$1$	$-4$	$1$	$1$	$0$	$1$	$-2$
102660.a.58e		$232$	$-8$	$4$	$2$	$4$	$2$	$-1$	$0$	$-1$	$-4$
102660.a.58f		$232$	$8$	$4$	$2$	$-4$	$-2$	$-1$	$0$	$1$	$-4$
102660.a.59a		$59$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$	$0$
102660.a.60a		$840$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$0$	$14$