C₂²×#616960 #72296049

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²³

Aliases

direct product

C₂²×#616960

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

3187256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72231761, C₂²×#72217875, C₂²×#72219674
maximal normal subgroup of	C₂²×#72231761, C₂²×#72217875, C₂²×#72219674
maximal quotient of	C₂²×#72218795

GAP

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

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

C22×#616960 #72296049