C₅×#8802 #11931522

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

Aliases

direct product

C₅×#8802

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#117
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#383
prime radical	C₂³×C₅	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

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

normal subgroups only

This group has 258 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) = 37
transitive degree	μ_t(G) = 320

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

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

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, 1790);