Abstract group 13063680.b: $\SU(4,3)$

• Label: 13063680.b
• Order: \( 2^{9} \cdot 3^{6} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 4
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{6} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $2160$
• Trans deg.: not computed
• Rank: $2$

Group information

Description:	$\SU(4,3)$
Order:	\(13063680\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 5 \cdot 7 \)
Exponent:	\(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Automorphism group:	$\PGammaU(4,3)$, of order \(26127360\)\(\medspace = 2^{10} \cdot 3^{6} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 2, $\PSU(4,3)$
Derived length:	$0$

This group is nonabelian and quasisimple (hence nonsolvable and perfect). Whether it is almost simple has not been computed.

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	12	14	18	20	28	36
Elements	1	5671	47600	141752	653184	501200	933120	2449440	483840	653184	1637440	933120	483840	1306368	1866240	967680	13063680
Conjugacy classes	1	2	4	7	1	8	2	5	4	1	16	2	4	2	4	8	71
Divisions	1	2	4	5	1	8	1	3	2	1	10	1	2	1	1	2	45
Autjugacy classes	1	2	3	5	1	6	1	3	1	1	8	1	1	1	2	1	38

Dimension	1	20	21	35	40	56	70	90	120	140	189	210	224	240	280	315	420	504	540	560	630	640	729	840	896	1080	1280	1680	1792	2560
Irr. complex chars.	1	3	1	2	0	2	5	1	3	3	1	3	8	0	10	2	3	2	3	2	1	8	1	2	4	0	0	0	0	0	71
Irr. rational chars.	1	1	1	2	1	2	1	1	1	3	1	1	0	1	1	2	2	2	1	7	1	0	1	1	4	1	2	1	1	1	45

Minimal presentations

Permutation degree:	$2160$
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	20	40	40
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\SU(4,3)$, $\SpinMinus(6,3)$
Permutation group:	Degree $2160$ $\langle(1,3,7,15,31,61,118,225,414)(2,5,11,23,42,82,157,294,536)(4,9,19,39,77,147,280,512,884) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\SU(4,3)$.

Homology

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

Subgroups

Subgroup data has not been computed.

Character theory

Complex character table

See the $71 \times 71$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $45 \times 45$ rational character table.