C₂²×#521402 #72339146

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²²

Aliases

direct product

C₂²×#521402

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁵×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1215032 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72337376
maximal normal subgroup of	C₂×#72337376

GAP

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

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

C22×#521402 #72339146