C₅×#7775 #11930495

order	1280
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7775

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

380 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 37
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 763);