C₅×#10629 #11933349

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

Aliases

direct product

C₅×#10629

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#397
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 310 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) = 45
transitive degree	μ_t(G) = 320

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

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

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, 3617);