C₅×#9976 #11932696

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9974
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#9976

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

716 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(6,7)(8,10)(9,11)(12,13)(14,15,20,24)(16,25,29,37)(17,30,34,40)(18,35,19,23)(21,28,38,27)(22,32,41,33)(26,43,44,36)(31,42,45,39);(10,13)(14,16)(15,21)(17,31)(18,28)(19,27)(20,29)(22,39)(23,37)(24,38)(25,35)(26,32)(30,36)(33,44)(34,45)(40,43)(41,42);(6,8)(7,10)(9,12)(11,13)(14,17,20,34)(15,22,24,41)(16,26,29,44)(18,32,19,33)(21,36,38,43)(23,40,35,30)(25,42,37,39)(27,31,28,45)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2964);