C₅×#8637 #11931357

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#8637

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

336 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 138 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)(6,7,11,14,12,9,10,20)(8,15,18,16,19,21,17,13)(23,27)(24,30)(25,33)(29,35)(31,32)(34,36);(6,8,10,17,12,19,11,18)(7,13,14,15,9,16,20,21)(22,23,25,27,26,29,33,35)(24,28,31,34,32,37,30,36);(6,9)(7,12)(8,16)(10,20)(11,14)(13,19)(15,18)(17,21)(22,24,26,32)(23,28,29,37)(25,31,33,30)(27,34,35,36)

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, 1625);