C₅×#9436 #11932156

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9431
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9436

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

400 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 640

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

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2424);