C₅×#10093 #11932813

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10088
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10093

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

496 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 102 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)(8,16)(9,12)(10,14)(11,15)(13,17)(18,20)(19,21)(22,23)(24,32)(25,29)(26,35)(27,31)(28,33)(30,36)(34,37);(7,12)(9,10)(13,16)(14,15)(17,19)(20,21)(22,24,27,25)(23,28,31,37)(26,34,36,33)(29,35,32,30);(6,8,11,18)(7,13,15,21)(9,17,10,19)(12,20,14,16)(22,25,27,24)(23,29,31,32)(26,33,36,34)(28,30,37,35)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3081);