C₂²×#519339 #72321297

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂²×#519339

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁴×#525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1036736 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);(10,12)(13,14,15,16);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,20,22)(19,21,24,23);(6,8)(17,18)(19,23)(20,22)(21,24);(1,2)(3,4);(17,19)(18,21)(20,24)(22,23);(5,6)(7,8)(17,19)(18,21)(20,24)(22,23)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72233877
maximal normal subgroup of	C₂²×#72233877
maximal quotient of	C₂²×#95×#120

GAP

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

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

C22×#519339 #72321297