Abstract group 16427999232000.a: $\GSp(4,16)$

• Label: 164...000.a
• Order: \( 2^{16} \cdot 3^{3} \cdot 5^{3} \cdot 17^{2} \cdot 257 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 17 \cdot 257 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 3 \cdot 5 \)
• $\card{Z(G)}$: 15
• $\card{\Aut(G)}$: \( 2^{22} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2} \cdot 257 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $4377$
• Trans deg.: not computed
• Rank: $2$

Group information

Description:	$\GSp(4,16)$
Order:	\(16427999232000\)\(\medspace = 2^{16} \cdot 3^{3} \cdot 5^{3} \cdot 17^{2} \cdot 257 \)
Exponent:	\(262140\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 17 \cdot 257 \)
Automorphism group:	Group of order \(70092796723200\)\(\medspace = 2^{22} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2} \cdot 257 \)
Composition factors:	$C_3$, $C_5$, $\Sp(4,16)$
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	10	12	15	17	20	30	34	51	60	85	102	170	255	257	510	771	1285	3855
Elements	1	16842495	107372546	4278124800	24695685124	27413683710	91334032380	8556249600	1512234909704	106361978880	17112499200	839788034040	64423526400	418879242240	34224998400	1112632197120	128847052800	257694105600	7172991221760	272734617600	515388211200	545469235200	1090938470400	2181876940800	16427999232000
Conjugacy classes	1	3	8	2	29	12	32	4	502	44	8	208	16	136	16	336	32	64	1824	64	128	128	256	512	4365
Divisions	1	3	5	2	9	7	9	2	73	6	2	29	2	10	2	12	2	2	32	1	2	1	1	1	216

Minimal presentations

Permutation degree:	$4377$
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:	$\GSp(4,16)$, $\GOrth(5,16)$
Permutation group:	Degree $4377$ $\langle(1,4,2,10,8,32,30,113,111,77,79,20,22,5,7)(3,14,12,48,46,166,164,543,541,204,206,57,59,15,16) \!\cdots\! \rangle$
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 $\GSp(4,16)$.

Homology

Abelianization:	$C_{15} \simeq C_{3} \times C_{5}$
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

Subgroup data has not been computed.

Character theory

Complex character table

The $4365 \times 4365$ character table is not available for this group.

Rational character table

The $216 \times 216$ rational character table is not available for this group.