C₂²×#516383 #72289167

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²³

Aliases

direct product

C₂²×#516383

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁵×#120	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3366712 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72232804, C₂²×#72217875
maximal normal subgroup of	C₂²×#72232804, C₂²×#72217875

GAP

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

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

C22×#516383 #72289167