C₂³×#648360 #72342779

order	4096
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁹×C₄

Aliases

direct product

C₂³×#648360

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁸
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

83626675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#120²
maximal normal subgroup of	C₂³×#120²

GAP

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

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

C23×#648360 #72342779