C₂²×#34273 #72278513

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂²×#34273

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#534	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

142781 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

352 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#7674648, C₂²×#7682069
maximal normal subgroup of	C₂²×#7674648, C₂²×#7682069
maximal quotient of	C₂²×#576607

GAP

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

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

C22×#34273 #72278513