C₅×#10495 #11933215

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10488
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10495

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

344 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 98 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)(8,11)(9,12)(10,14)(13,18)(15,19)(16,21)(17,20)(22,23,27,31)(24,32,33,36)(25,28,26,30)(29,34,37,35);(6,8,10,16)(7,11,14,21)(9,17,18,15)(12,20,13,19)(23,28)(24,33)(25,26)(29,32)(30,31)(36,37);(7,12)(9,18)(11,19)(13,14)(15,17)(20,21)(22,24,25,34,27,33,26,35)(23,29,30,36,31,37,28,32)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3483);