Abstract group 28569600.a: $\GUnitary(2,31)$

• Label: 28569600.a
• Order: \( 2^{12} \cdot 3^{2} \cdot 5^{2} \cdot 31 \)
• Exponent: \( 2^{6} \cdot 3 \cdot 5 \cdot 31 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \cdot 3 \cdot 5 \)
• $\card{Z(G)}$: 960
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3 \cdot 5 \cdot 31 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $2056$
• Trans deg.: not computed
• Rank: $2$

Group information

Description:	$\GUnitary(2,31)$
Order:	\(28569600\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 5^{2} \cdot 31 \)
Exponent:	\(29760\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 31 \)
Automorphism group:	Group of order \(15237120\)\(\medspace = 2^{15} \cdot 3 \cdot 5 \cdot 31 \)
Composition factors:	$C_2$ x 7, $C_3$, $C_5$, $\PSL(2,31)$
Derived length:	$1$

This group is nonabelian and nonsolvable. Whether it is almost simple has not been computed.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	16	20	24	30	31	32	40	48	60	62	64	80	93	96	120	124	155	160	186	192	240	248	310	320	372	465	480	496	620	744	930	960	992	1240	1488	1860	1984	2480	2976	3720	4960	5952	7440	9920	14880	29760
Elements	1	1923	2978	5644	9924	12774	22448	37452	23192	91272	89536	62256	68704	289176	960	357632	169152	226688	410208	960	477184	516864	1920	810496	909696	1920	3840	1747968	1920	1144832	2176512	3840	3840	2543616	3840	7680	5781504	7680	7680	7680	7680	9658368	15360	15360	15360	15360	30720	30720	30720	30720	61440	61440	61440	122880	122880	245760	28569600
Conjugacy classes	1	3	5	8	14	15	28	42	28	100	104	72	80	300	1	400	192	256	432	1	544	576	2	896	960	2	4	1920	2	1280	2304	4	4	2816	4	8	6144	8	8	8	8	10240	16	16	16	16	32	32	32	32	64	64	64	128	128	256	30720
Divisions	1	3	3	5	4	8	9	11	9	14	15	11	13	39	1	27	15	19	31	1	19	21	1	31	35	1	1	33	1	23	41	1	1	25	1	1	53	1	1	1	1	45	1	1	1	1	1	1	1	1	1	1	1	1	1	1	591
Autjugacy classes	1	3	3	4	5	7	8	11	8	23	16	12	12	47	1	34	16	20	48	1	50	24	1	38	52	1	1	42	1	54	60	1	1	58	1	1	78	1	1	1	1	94	1	1	1	1	1	1	1	1	1	1	1	1	1	1	856

Minimal presentations

Permutation degree:	$2056$
Transitive degree:	not computed
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Groups of Lie type:	$\GUnitary(2,31)$
Permutation group:	Degree $2056$ $\langle(1,3,6)(2,9,11)(4,13,17)(5,18,20)(7,14,24)(8,27,29)(10,32,36)(12,34,39) \!\cdots\! \rangle$
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 $\GUnitary(2,31)$.

Homology

Abelianization:	$C_{2} \times C_{480} \simeq C_{2} \times C_{32} \times C_{3} \times C_{5}$
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

Subgroup data has not been computed.

Character theory

Complex character table

The $30720 \times 30720$ character table is not available for this group.

Rational character table

The $591 \times 591$ rational character table is not available for this group.