C₅×#13315 #11936035

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13315

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

516 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 73
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 73 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6303);