Abstract group 32537600.a: ${}^2B(2,32)$

• Label: 32537600.a
• Order: \( 2^{10} \cdot 5^{2} \cdot 31 \cdot 41 \)
• Exponent: \( 2^{2} \cdot 5^{2} \cdot 31 \cdot 41 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{10} \cdot 5^{3} \cdot 31 \cdot 41 \)
• $\card{\mathrm{Out}(G)}$: \( 5 \)
• Perm deg.: $1025$
• Trans deg.: not computed
• Rank: $2$

Group information

Description:	${}^2B(2,32)$
Order:	\(32537600\)\(\medspace = 2^{10} \cdot 5^{2} \cdot 31 \cdot 41 \)
Exponent:	\(127100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 31 \cdot 41 \)
Automorphism group:	${}^2B(2,32).C_5$, of order \(162688000\)\(\medspace = 2^{10} \cdot 5^{3} \cdot 31 \cdot 41 \)
Composition factors:	${}^2B(2,32)$
Derived length:	$0$

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

Group statistics

Order	1	2	4	5	25	31	41
Elements	1	31775	1016800	1301504	6507520	15744000	7936000	32537600
Conjugacy classes	1	1	2	1	5	15	10	35
Divisions	1	1	1	1	1	1	1	7
Autjugacy classes	1	1	2	1	1	3	2	11

Dimension	1	124	248	775	1024	1025	1271	6355	7750	15375
Irr. complex chars.	1	2	0	10	1	15	6	0	0	0	35
Irr. rational chars.	1	0	1	0	1	0	1	1	1	1	7

Minimal presentations

Permutation degree:	$1025$
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	124	248	248
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $1025$ $\langle(2,3,7,4)(5,13,40,15)(6,17,53,18)(8,22,70,24)(9,26,83,28)(10,30,96,31)(11,32,103,34) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ \alpha^{4} & 1 & 0 & 0 \\ 0 & \alpha & 1 & 0 \\ \alpha^{9} & \alpha^{5} & \alpha^{4} & 1 \\ \end{array}\right), \left(\begin{array}{llll}\alpha^{4} & 0 & 0 & 0 \\ 0 & \alpha^{28} & 0 & 0 \\ 0 & 0 & \alpha^{3} & 0 \\ 0 & 0 & 0 & \alpha^{27} \\ \end{array}\right), \left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{32}) = \GL_{4}(\F_{2}[\alpha]/(\alpha^{5} + \alpha^{2} + 1))$
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 $\GL_{4}(\F_{32})$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

Subgroup data has not been computed.

Character theory

Complex character table

See the $35 \times 35$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	4A	5A	25A	31A	41A
	Size	1	31775	1016800	1301504	6507520	15744000	7936000
	2 P	1A	1A	2A	5A	25A	31A	41A
	5 P	1A	2A	4A	5A	25A	31A	41A
	31 P	1A	2A	4A	1A	5A	31A	41A
	41 P	1A	2A	4A	5A	25A	1A	41A
32537600.a.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$
32537600.a.124a		$248$	$-8$	$0$	$-2$	$-2$	$0$	$2$
32537600.a.775a		$7750$	$70$	$-10$	$0$	$0$	$0$	$1$
32537600.a.1024a		$1024$	$0$	$0$	$-1$	$-1$	$1$	$-1$
32537600.a.1025a		$15375$	$15$	$15$	$0$	$0$	$-1$	$0$
32537600.a.1271a		$1271$	$-9$	$-1$	$-4$	$1$	$0$	$0$
32537600.a.1271b		$6355$	$-45$	$-5$	$5$	$0$	$0$	$0$