C₂²×#35240 #72339115

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂²×#35240

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

108037 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7628492, C₂²×#7737454
maximal normal subgroup of	C₂²×#7628492, C₂²×#7737454
maximal quotient of	C₂²×#613325, C₂²×#517516

GAP

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

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

C22×#35240 #72339115