C₅×#9446 #11932166

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9443
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₅×#9446

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂×C₅×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

516 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7)(8,14)(9,17,19,12)(10,16)(11,13)(15,18,21,20)(22,23)(24,27,31,34)(25,28)(26,29)(30,33,36,37)(32,35);(6,8,14,7,11,16,10,13)(9,15,18,12,19,21,20,17)(22,24,26,31)(23,27)(25,30,32,36)(28,33)(29,34)(35,37);(6,9,11,19)(7,12,13,17)(8,15,16,21)(10,20,14,18)(22,25,26,32)(23,28,29,35)(24,30,31,36)(27,33,34,37)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2434);