C₅×#10630 #11933350

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#10630

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#397
prime radical	C₂³×C₅	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 272 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) = 53
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7)(8,11)(9,12)(10,13)(14,21)(15,19)(16,18)(17,20)(22,23)(24,32,35,45)(25,29)(26,30)(27,31)(28,34,42,39)(33,48,36,47)(37,43)(38,44)(40,46)(41,50,52,49)(51,53);(6,8,9,15)(7,11,12,19)(10,14,17,16)(13,21,20,18)(22,24)(23,28)(25,36)(26,39)(27,35)(29,41)(30,45)(31,42)(32,46)(33,38)(34,40)(37,49)(43,48)(44,52)(47,53)(50,51);(7,12)(8,14)(10,17)(11,18)(15,16)(19,21)(22,25,26,37,27,38,40,51)(23,29,30,43,31,44,46,53)(24,33,34,49,35,36,39,50)(28,41,32,47,42,52,45,48)

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, 3618);