C₂²×#14418 #72337029

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂²×#14418

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

82189 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#846769, C₂²×#531763, C₂²×#846753, …
maximal normal subgroup of	C₂²×#846769, C₂²×#531763, C₂²×#846753, …

GAP

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

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

C22×#14418 #72337029