C₅×#10670 #11933390

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#10670

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#399
prime radical	C₂³×C₅	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

316 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(6,7,9,11,10,13,8,12)(14,15,20,25,19,24,18,23)(16,26,35,21,29,37,28,38)(17,30,34,40,33,22,32,41)(27,39,31,36,43,44,42,45);(7,11)(8,9)(12,13)(14,16,18,35,19,29,20,28)(15,21,23,37,24,38,25,26)(17,31,32,43,33,42,34,27)(22,39,40,36,30,44,41,45);(6,7,10,13)(8,11,9,12)(14,17,19,33)(15,22,24,30)(16,27,29,43)(18,32,20,34)(21,36,38,45)(23,40,25,41)(26,39,37,44)(28,42,35,31)

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, 3658);