Abstract group 53443952640000.a: $\PSU(5,4)$

• Label: 534...000.a
• Order: \( 2^{20} \cdot 3^{2} \cdot 5^{4} \cdot 13 \cdot 17 \cdot 41 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \cdot 41 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{22} \cdot 3^{2} \cdot 5^{5} \cdot 13 \cdot 17 \cdot 41 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 5 \)
• Perm deg.: $17425$
• Trans deg.: $17425$
• Rank: $2$

Group information

Description:	$\PSU(5,4)$
Order:	\(53443952640000\)\(\medspace = 2^{20} \cdot 3^{2} \cdot 5^{4} \cdot 13 \cdot 17 \cdot 41 \)
Exponent:	\(1087320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \cdot 41 \)
Automorphism group:	Group of order \(1068879052800000\)\(\medspace = 2^{22} \cdot 3^{2} \cdot 5^{5} \cdot 13 \cdot 17 \cdot 41 \)
Composition factors:	$\PSU(5,4)$
Derived length:	$0$

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

Group statistics

Order	1	2	3	4	5	6	8	10	12	13	15	17	20	26	30	39	41	51	65
Elements	1	13643775	4651827200	55670784000	91791130624	1169086464000	1043827200000	1581050265600	1113415680000	274071552000	4037987532800	4191682560000	1336098816000	4111073280000	3562930176000	5481431040000	10428088320000	8383365120000	6577717248000	53443952640000
Conjugacy classes	1	2	2	3	5	2	5	8	1	4	8	4	2	4	4	4	8	8	8	83
Divisions	1	2	2	3	3	2	5	3	1	1	3	1	1	1	1	1	1	1	1	34
Autjugacy classes	1	2	2	3	3	2	1	3	1	1	3	2	1	1	1	1	2	2	2	34

Minimal presentations

Permutation degree:	$17425$
Transitive degree:	$17425$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	204	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PSU(5,4)$
Permutation group:	Degree $17425$ $\langle(1,2,3,4,7,14,33,83,219,580,1499,1609,425,1106,2798,6558,10219,13813,16103,8367,14207,17114,5228,10138,13680,1415,3544,7952,6746,12642,7290,13195,9287,12364,2954,529,1368,3431,7746,7737,12668,6379,12226,5528,8030,6999,1710,4194,9024,14730,17201) \!\cdots\! \rangle$
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 $\SU(5,4)$.

Homology

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

Subgroups

There are 2 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $\PSU(5,4)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.C_2^6.C_2^6.C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^3:C_5$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$
41-Sylow subgroup:	$P_{ 41 } \simeq$ $C_{41}$

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

All subgroups of index up to 18000 (order at least 2969108480) are shown, as well as all normal subgroups of any index.

Order 53443952640000: $\PSU(5,4)$

Order 3067084800: $C_2^2.C_2^{12}.C_3.\SU(3,4)$

Order 1018368000: $\SU(4,4)$

Order 802160640: $C_2^8.C_2^8.C_3.\SL(2,16)$

Order 3744000: $A_5\times \SU(3,4)$

Order 1048576: $C_2^6.C_2^6.C_2^6.C_2^2$

Order 15000: $C_5^3:S_5$

Order 3000: $C_5^2.\SL(2,5)$ x 5

Order 625: $C_5^3:C_5$

Order 205: $C_{41}:C_5$

Order 41: $C_{41}$

Order 17: $C_{17}$

Order 13: $C_{13}$

Order 9: $C_3^2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 53443952640000: $\PSU(5,4)$ ($C_1$)

Order 1: $C_1$ ($\PSU(5,4)$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

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

The $83 \times 83$ character table is not available for this group.

Rational character table

The $34 \times 34$ rational character table is not available for this group.