Abstract group 34693789777920.a: $\SU(4,8)$

• Label: 346...920.a
• Order: \( 2^{18} \cdot 3^{7} \cdot 5 \cdot 7^{2} \cdot 13 \cdot 19 \)
• Exponent: \( 2^{2} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 13 \cdot 19 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{19} \cdot 3^{8} \cdot 5 \cdot 7^{2} \cdot 13 \cdot 19 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $4617$
• Trans deg.: not computed
• Rank: $2$

Group information

Description:	$\SU(4,8)$
Order:	\(34693789777920\)\(\medspace = 2^{18} \cdot 3^{7} \cdot 5 \cdot 7^{2} \cdot 13 \cdot 19 \)
Exponent:	\(933660\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 13 \cdot 19 \)
Automorphism group:	Group of order \(208162738667520\)\(\medspace = 2^{19} \cdot 3^{8} \cdot 5 \cdot 7^{2} \cdot 13 \cdot 19 \)
Composition factors:	$\SU(4,8)$
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	7	9	12	13	14	18	19	21	27	35	36	42	57	63	65	91	126	171	455	513
Elements	1	17039295	865479680	68702437440	76250087424	117357327360	239346855936	689265607680	120464547840	228750262272	438835138560	1637027159040	405775319040	1265287495680	405775319040	457500524544	361393643520	413021306880	811550638080	8201423093760	915001049088	1372501573632	1239063920640	2434651914240	5490006294528	7303955742720	34693789777920
Conjugacy classes	1	2	4	2	1	6	9	50	2	3	6	42	6	24	6	6	6	6	12	144	12	18	18	36	72	108	602
Divisions	1	2	3	2	1	4	3	11	1	1	2	8	1	5	1	1	1	1	1	9	1	1	1	1	1	1	65
Autjugacy classes	1	2	3	2	1	4	3	11	1	1	2	8	1	5	1	1	1	1	2	27	4	3	3	6	12	18	124

Minimal presentations

Permutation degree:	$4617$
Transitive degree:	not computed
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	455	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\SU(4,8)$, $\PSU(4,8)$, $\OmegaMinus(6,8)$, $\PGU(4,8)$, $\POmegaMinus(6,8)$, $\SpinMinus(6,8)$, $\CSU(4,8)$
Permutation group:	Degree $4617$ $\langle(1,3,7,15,31,63)(2,5,11,23,47,94)(4,9,19,39,72,142)(6,13,27,55,110,215) \!\cdots\! \rangle$
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 matrices in $\SU(4,8)$.

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 $\SU(4,8)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^{12}.C_2^6$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9\wr C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}$

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 1755 (order at least 19768541184) are shown, as well as all normal subgroups of any index.

Order 34693789777920: $\SU(4,8)$

Order 7514357760: $C_2^{12}.C_7.\SL(2,64)$

Order 1056706560: $\Sp(4,8)$

Order 1040449536: $C_2^3.C_2^{12}.C_{63}.\SL(2,8)$

Order 148925952: $\Unitary(3,8)$

Order 4572288: $C_9.\SL(2,8)^2.C_2$

Order 3669120: $C_7.\SL(2,64).C_2$

Order 262144: $C_2^{12}.C_2^6$

Order 25920: $\SU(4,2)$

Order 17496: $C_9^3:S_4$

Order 2187: $C_9\wr C_3$

Order 49: $C_7^2$

Order 19: $C_{19}$

Order 13: $C_{13}$

Order 5: $C_5$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 34693789777920: $\SU(4,8)$ ($C_1$)

Order 1: $C_1$ ($\SU(4,8)$)

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

Character theory

Complex character table

The $602 \times 602$ character table is not available for this group.

Rational character table

The $65 \times 65$ rational character table is not available for this group.