C₅×#12168 #11934888

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12168

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#424 ◁ G

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

normal subgroups only

This group has 330 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) = 77
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 77 points:

(1,2,3,4,5)(10,13)(14,15,21,27)(16,28,32,56)(17,33,36,53)(18,37,41,51)(19,26,20,25)(22,44,47,30)(23,42,49,35)(24,39,52,40)(29,57,61,74)(31,62,43,50)(34,65,68,48)(38,54,69,46)(45,66,73,77)(55,60,76,59)(58,63,71,72)(64,70,75,67);(6,7)(8,10)(9,11)(12,13)(14,16,21,32)(15,22,27,47)(17,29,36,61)(18,38,41,69)(19,31,20,43)(23,45,49,73)(24,50,52,62)(25,46,26,54)(28,51,56,37)(30,39,44,40)(33,63,53,72)(34,66,68,77)(35,59,42,60)(48,74,65,57)(55,75,76,64)(58,70,71,67);(6,8,9,12)(7,10,11,13)(14,17)(15,23)(16,29)(18,34)(19,42)(20,35)(21,36)(22,45)(24,48)(25,53)(26,33)(27,49)(28,55)(30,58)(31,60)(32,61)(37,64)(38,66)(39,70)(40,67)(41,68)(43,59)(44,71)(46,72)(47,73)(50,74)(51,75)(52,65)(54,63)(56,76)(57,62)(69,77)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5156);