C₂²×#7646026 #72337201

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#7646026

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1623352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 24 points:

(9,10,13,14)(11,12,15,16);(5,6)(7,8)(10,14)(11,15)(17,18,21,22)(19,20,23,24);(5,6)(7,8)(9,10,13,14)(11,16,15,12)(17,19,21,23)(18,20,22,24);(6,8)(17,19)(18,20)(21,23)(22,24);(3,4);(9,11)(10,12)(13,15)(14,16);(1,2)(3,4);(5,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,20)(18,19)(21,24)(22,23)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72235095
maximal normal subgroup of	C₂²×#72235095

GAP

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

gap> G := Group((9,10,13,14)(11,12,15,16), (5,6)(7,8)(10,14)(11,15)(17,18,21,22)(19,20,23,24), (5,6)(7,8)(9,10,13,14)(11,16,15,12)(17,19,21,23)(18,20,22,24), (6,8)(17,19)(18,20)(21,23)(22,24), (3,4), (9,11)(10,12)(13,15)(14,16), (1,2)(3,4), (5,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,20)(18,19)(21,24)(22,23));

C22×#7646026 #72337201