C₂²×#7670206 #72296010

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

Aliases

direct product

C₂²×#7670206

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

3435416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72217511, C₂²×#72232065
maximal normal subgroup of	C₂²×#72217511, C₂²×#72232065

GAP

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

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

C22×#7670206 #72296010