C₂²×#37703 #72324830

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#37703

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

104689 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)(10,14)(11,16)(18,21)(22,24);(9,10,13,14)(11,12,16,15)(18,21)(19,23);(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24);(3,4);(6,8)(9,12,13,15)(10,11,14,16)(17,19)(18,22)(20,23)(21,24);(1,2)(3,4);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7712649
maximal normal subgroup of	C₂²×#7712649

GAP

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

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

C22×#37703 #72324830