C₅×#11670 #11934390

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11670

Subgroups

	subgroup	G/H
center	C₄×C₅	#418
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂×C₅	#1490
prime radical	C₅×#575	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₄×C₅×M₁₆ ◁ G

348 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 160

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

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4658);