Abstract group 1165572172800.a: $\PSU(4,7)$

• Label: 116...800.a
• Order: \( 2^{10} \cdot 3^{2} \cdot 5^{2} \cdot 7^{6} \cdot 43 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 43 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{2} \cdot 5^{2} \cdot 7^{6} \cdot 43 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $2752$
• Trans deg.: $2752$
• Rank: $2$

Group information

Description:	$\PSU(4,7)$
Order:	\(1165572172800\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5^{2} \cdot 7^{6} \cdot 43 \)
Exponent:	\(361200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 43 \)
Automorphism group:	Group of order \(9324577382400\)\(\medspace = 2^{13} \cdot 3^{2} \cdot 5^{2} \cdot 7^{6} \cdot 43 \)
Composition factors:	$\PSU(4,7)$
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	8	12	14	15	16	21	24	25	28	42	43	48	56	75	84	86
Elements	1	2683975	295687952	2498480600	15540962304	20524708400	13841287200	27173557600	28908040000	18300559200	31081924608	48565507200	29733984000	113319516800	77704811520	22796054400	13875859200	189744307200	97131014400	41627577600	155409623040	27751718400	189744307200	1165572172800
Conjugacy classes	1	2	2	3	1	3	8	7	4	5	2	4	3	8	5	5	1	14	8	4	10	2	14	116
Divisions	1	2	2	3	1	3	6	5	3	5	1	1	3	4	1	4	1	1	1	2	1	1	1	53
Autjugacy classes	1	2	2	3	1	3	4	6	3	4	1	2	2	6	5	3	1	7	4	3	5	1	7	76

Minimal presentations

Permutation degree:	$2752$
Transitive degree:	$2752$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	300	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PSU(4,7)$, $\POmegaMinus(6,7)$
Permutation group:	Degree $2752$ $\langle(1,3,7,15,31,63,126,245,461,603,1069,1751,2402,250,470,853,1448,2164,1706,1040,1711) \!\cdots\! \rangle$
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 $\SU(4,7)$.

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 $\PSU(4,7)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.C_2^4.C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^4.C_7^2$
43-Sylow subgroup:	$P_{ 43 } \simeq$ $C_{43}$

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 158025 (order at least 7375872) are shown, as well as all normal subgroups of any index.

Order 1165572172800: $\PSU(4,7)$

Order 423536400: $C_7^4.C_3.\PSL(2,49)$

Order 141178800: $C_7^4.\PSL(2,49)$

Order 138297600: $\PSp(4,7)$ x 2

Order 67765824: $C_7.C_7^4.C_{12}.\SO(3,7)$

Order 33882912: $C_7.C_7^4.C_{12}.\PSL(2,7)$

Order 22588608: $C_7.C_7^4.C_4.\SO(3,7)$

Order 16941456: $\He_7.C_7^2.C_3.\SL(2,7)$

Order 11327232: $C_2\times \SU(3,7)$

Order 11294304: $C_7.C_7^4.C_4.\PSL(2,7)$

Order 8470728: $C_7^3.C_7^3.C_3.C_{24}$

Order 451584: $C_4.\PSL(2,7)^2.C_2^2$

Order 352800: $C_3.\PSL(2,49).C_2$

Order 117649: $C_7^4.C_7^2$

Order 117600: $\POMinus(4,7)$ x 2

Order 112896: $\POPlus(4,7)$ x 2

Order 11520: $C_2^4.S_6$ x 4

Order 3072: $(C_2\times C_8^2).S_4$

Order 1024: $C_4^2.C_2^4.C_2^2$

Order 43: $C_{43}$

Order 25: $C_{25}$

Order 9: $C_3^2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 1165572172800: $\PSU(4,7)$ ($C_1$)

Order 1: $C_1$ ($\PSU(4,7)$)

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 $116 \times 116$ character table is not available for this group.

Rational character table

The $53 \times 53$ rational character table is not available for this group.