C₂²×#759292 #72324214

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²²×C₄

Aliases

direct product

C₂²×#759292

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1656440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×D₄×#2403
maximal normal subgroup of	C₂²×D₄×#2403

GAP

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

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

C22×#759292 #72324214