C₅×#10668 #11933388

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

Aliases

direct product

C₅×#10668

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

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, 3656);