C₅×#8799 #11931519

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#8799

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#382
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

220 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 640

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

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

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, 1787);