C₂²×#158317 #72322019

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂²×#158317

Subgroups

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

1087456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72278330, C₂²×#72278357
maximal normal subgroup of	C₂²×#72278330, C₂²×#72278357

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

C22×#158317 #72322019