C₅×#8782 #11931502

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#8782

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#117
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#382
prime radical	C₂³×C₅	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#376 ◁ G

262 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 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)(6,7,9,12,10,13,16,19)(8,11,15,18,14,17,20,21)(22,23)(24,32,35,28)(25,36,38,51)(26,31)(27,30)(29,37,42,52)(33,46)(34,45,49,41)(39,43)(40,47)(44,53)(48,50);(6,8)(7,11)(9,15)(10,14)(12,18)(13,17)(16,20)(19,21)(22,24,26,34,27,35,39,49)(23,28,31,41,30,32,43,45)(25,33,37,47,38,48,52,53)(29,40,36,50,42,44,51,46);(8,14)(11,17)(15,20)(18,21)(22,25)(23,29,30,42)(24,33)(26,37)(27,38)(28,40,32,44)(31,36,43,51)(34,47)(35,48)(39,52)(41,50,45,46)(49,53)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1770);