C₂²×#14300 #72337043

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂²×#14300

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

77917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 24 points:

(6,7)(9,10)(11,12)(17,18,21,24)(19,22,23,20);(10,11)(19,23)(20,22);(5,6,8,7)(9,10,12,11)(13,14,15,16)(17,19)(18,22)(20,24)(21,23);(3,4);(5,7,8,6)(9,11,12,10)(17,20,21,22)(18,23,24,19);(1,2)(3,4)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#1259290, C₂²×#1259282, C₂²×#1393170, …
maximal normal subgroup of	C₂²×#1259290, C₂²×#1259282, C₂²×#1393170, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((6,7)(9,10)(11,12)(17,18,21,24)(19,22,23,20), (10,11)(19,23)(20,22), (5,6,8,7)(9,10,12,11)(13,14,15,16)(17,19)(18,22)(20,24)(21,23), (3,4), (5,7,8,6)(9,11,12,10)(17,20,21,22)(18,23,24,19), (1,2)(3,4));

C22×#14300 #72337043