C₂²×#35346 #72232030

order	1024
nilpotency class	2
isoclinism	isoclinic class of #35346
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²²

Aliases

direct product

C₂²×#35346

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

438421 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 20 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂²×#35346⋊C₃, #72258418, #72258456, …
maximal subgroup of	#72241126, C₂²×#245225, C₂²×#35346⋊C₃, …
maximal normal subgroup of	#72241126, C₂²×#245225, C₂²×#35346⋊C₃, …
maximal quotient of	#72232064, C₂×#72217875, C₂²×#6499874, …

GAP

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

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

C22×#35346 #72232030