C₂²×#4928753 #72321631

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

direct product

C₂²×#4928753

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

831888 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72279977
maximal normal subgroup of	C₂²×#72279977
maximal quotient of	C₂²×#72278330

GAP

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

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

C22×#4928753 #72321631