C₅×#10373 #11933093

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10373

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#393
prime radical	C₂³×C₅	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

364 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 106 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)(8,15,17,18)(12,16,20,21)(22,23,25,27)(24,30,31,37)(26,34,33,29)(28,35,36,32);(6,7)(8,16,17,21)(9,19)(10,13)(11,14)(12,18,20,15)(23,27)(28,36)(29,34)(30,37);(6,8)(7,12)(9,18)(10,15)(11,17)(13,21)(14,20)(16,19)(22,24,25,31)(23,28,27,36)(26,35,33,32)(29,37,34,30)

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, 3361);