C₅×#12437 #11935157

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12437

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1667
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ G

214 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,7,14,21)(8,22,28,52)(9,29,32,60)(10,18,34,46)(11,17,33,47)(12,20,13,19)(15,37,42,24)(16,43,45,30)(23,53,56,64)(25,41,57,40)(26,51,27,39)(31,61,36,48)(35,49,62,44)(38,55,65,59)(50,63,67,54)(58,68,69,66);(6,8,11,25,12,26,34,37,14,28,33,57,13,27,10,24)(7,15,18,39,19,40,47,52,21,42,46,51,20,41,17,22)(9,23,35,58,31,55,43,63,32,56,62,69,36,59,30,54)(16,38,48,66,44,64,60,67,45,65,61,68,49,53,29,50);(6,9)(7,16)(8,23)(10,30)(11,35)(12,31)(13,36)(14,32)(15,38)(17,29)(18,48)(19,44)(20,49)(21,45)(22,50)(24,54)(25,58)(26,55)(27,59)(28,56)(33,62)(34,43)(37,63)(39,66)(40,64)(41,53)(42,65)(46,61)(47,60)(51,68)(52,67)(57,69)

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, 5425);