Abstract group $\PSL(2,73)$

• Label: 194472.a
• Order: \( 2^{3} \cdot 3^{2} \cdot 37 \cdot 73 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 37 \cdot 73 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{4} \cdot 3^{2} \cdot 37 \cdot 73 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $74$
• Trans deg.: $74$
• Rank: $2$

Group information

Description:	$\PSL(2,73)$
Order:	\(194472\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 37 \cdot 73 \)
Exponent:	\(97236\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \cdot 73 \)
Automorphism group:	$\PGL(2,73)$, of order \(388944\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 37 \cdot 73 \)
Outer automorphisms:	$C_2$, of order \(2\)
Composition factors:	$\PSL(2,73)$
Derived length:	$0$

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

Group statistics

Order	1	2	3	4	6	9	12	18	36	37	73
Elements	1	2701	5402	5402	5402	16206	10804	16206	32412	94608	5328	194472
Conjugacy classes	1	1	1	1	1	3	2	3	6	18	2	39
Divisions	1	1	1	1	1	1	1	1	1	1	1	11
Autjugacy classes	1	1	1	1	1	3	2	3	6	18	1	38

Dimension	1	37	72	73	74	148	222	444	1296
Irr. complex chars.	1	2	18	1	17	0	0	0	0	39
Irr. rational chars.	1	0	0	1	4	1	2	1	1	11

Minimal Presentations

Permutation degree:	$74$
Transitive degree:	$74$
Rank:	$2$
Inequivalent generating pairs:	$95856$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	37	37	73
Arbitrary	37	37	73

Constructions

Groups of Lie type:	$\PSL(2,73)$
Permutation group:	Degree $74$ $\langle(3,67,43,70,39,31,50,9,47,29,51,12,52,45,23,57,42,71,60,44,16,14,11,24,40,58,5,15,21,41,28,64,18,55,4,8) \!\cdots\! \rangle$
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 $\SL(2,73)$.

Homology

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

Subgroups

There are 176087 subgroups in 38 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\PSL(2,73)$
Commutator:	$G' \simeq$ $\PSL(2,73)$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\PSL(2,73)$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\PSL(2,73)$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\PSL(2,73)$
Socle:	$\operatorname{soc} \simeq$ $\PSL(2,73)$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
37-Sylow subgroup:	$P_{ 37 } \simeq$ $C_{37}$
73-Sylow subgroup:	$P_{ 73 } \simeq$ $C_{73}$

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 194472: $\PSL(2,73)$

Order 2628: $C_{73}:C_{36}$

Order 1314: $C_{73}:C_{18}$

Order 876: $C_{73}:C_{12}$

Order 657: $C_{73}:C_9$

Order 438: $C_{73}:C_6$

Order 292: $C_{73}:C_4$

Order 219: $C_{73}:C_3$

Order 146: $D_{73}$

Order 74: $D_{37}$

Order 73: $C_{73}$

Order 72: $D_{36}$

Order 37: $C_{37}$

Order 36: $D_{18}$ x 2, $C_{36}$

Order 24: $S_4$ x 2, $D_{12}$

Order 18: $D_9$ x 2, $C_{18}$

Order 12: $D_6$ x 2, $A_4$ x 2, $C_{12}$

Order 9: $C_9$

Order 8: $D_4$

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

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 194472: $\PSL(2,73)$

Order 2628: $C_{73}:C_{36}$

Order 1314: $C_{73}:C_{18}$

Order 876: $C_{73}:C_{12}$

Order 657: $C_{73}:C_9$

Order 438: $C_{73}:C_6$

Order 292: $C_{73}:C_4$

Order 219: $C_{73}:C_3$

Order 146: $D_{73}$

Order 74: $D_{37}$

Order 73: $C_{73}$

Order 72: $D_{36}$

Order 37: $C_{37}$

Order 36: $D_{18}$, $C_{36}$

Order 24: $D_{12}$, $S_4$

Order 18: $C_{18}$, $D_9$

Order 12: $D_6$, $A_4$, $C_{12}$

Order 9: $C_9$

Order 8: $D_4$

Order 6: $C_6$, $S_3$

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

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 194472: $\PSL(2,73)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,73)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 194472: $\PSL(2,73)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,73)$)

Series

Derived series	$\PSL(2,73)$
Chief series	$\PSL(2,73)$	$\rhd$	$C_1$
Lower central series	$\PSL(2,73)$
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 $39 \times 39$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	4A	6A	9A	12A	18A	36A	37A	73A
	Size	1	2701	5402	5402	5402	16206	10804	16206	32412	94608	5328
	2 P	1A	1A	3A	2A	3A	9A	6A	9A	18A	37A	73A
	3 P	1A	2A	1A	4A	2A	3A	4A	6A	12A	37A	73A
	37 P	1A	2A	3A	4A	6A	9A	12A	18A	36A	1A	73A
	73 P	1A	2A	3A	4A	6A	9A	12A	18A	36A	37A	1A
194472.a.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
194472.a.37a		$74$	$2$	$2$	$-2$	$2$	$2$	$-2$	$2$	$-2$	$0$	$1$
194472.a.72a		$1296$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$-18$
194472.a.73a		$73$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$0$
194472.a.74a		$74$	$2$	$2$	$2$	$2$	$-1$	$2$	$-1$	$-1$	$0$	$1$
194472.a.74b		$74$	$-2$	$2$	$0$	$-2$	$2$	$0$	$-2$	$0$	$0$	$1$
194472.a.74c		$74$	$2$	$2$	$-2$	$2$	$-1$	$-2$	$-1$	$1$	$0$	$1$
194472.a.74d		$148$	$-4$	$4$	$0$	$-4$	$-2$	$0$	$2$	$0$	$0$	$2$
194472.a.74e		$222$	$6$	$-3$	$6$	$-3$	$0$	$-3$	$0$	$0$	$0$	$3$
194472.a.74f		$222$	$6$	$-3$	$-6$	$-3$	$0$	$3$	$0$	$0$	$0$	$3$
194472.a.74g		$444$	$-12$	$-6$	$0$	$6$	$0$	$0$	$0$	$0$	$0$	$6$