C₅×#7785 #11930505

order	1280
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#7785

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

454 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 94 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)(10,13)(15,18)(16,20)(19,21)(22,23,27,30)(24,31,33,36)(25,29,26,34)(28,35,37,32);(6,7,9,11)(8,10,12,13)(16,20)(18,21)(22,24,27,33)(23,28,30,37)(25,32,26,35)(29,36,34,31);(6,8,9,12)(7,10,11,13)(14,15,17,19)(16,21,20,18)(22,25,27,26)(23,29,30,34)(24,32,33,35)(28,36,37,31)

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, 773);