C₅×#10346 #11933066

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10346

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	#120
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ G

500 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 122 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)(6,7,9,11)(8,12,13,10)(14,15,16,18)(17,19,21,20)(22,23)(24,27)(25,28)(26,29)(30,35)(31,34)(32,36)(33,37);(6,8,9,13)(7,10,11,12)(15,18)(19,20)(22,24)(23,27)(25,30)(26,31)(28,35)(29,34)(32,37)(33,36);(15,19)(17,21)(18,20)(22,25,26,32)(23,28,29,36)(24,30,31,37)(27,33,34,35)

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, 3334);