C₂²×#17068 #72320795

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂²×#17068

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

87933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 20 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#822347, C₂²×#1409001, C₂²×#719412, …
maximal normal subgroup of	C₂²×#822347, C₂²×#1409001, C₂²×#719412, …
maximal quotient of	C₂²×#158362, C₂²×#158361

GAP

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

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

C22×#17068 #72320795