C₅×#9982 #11932702

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9980
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#9982

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

722 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 98 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)(14,15,20,25)(16,26,29,37)(17,22,33,40)(18,24,19,23)(21,30,36,28)(27,42,43,38)(31,45,44,35)(32,41,34,39);(10,13)(14,16)(15,21)(17,31)(18,30)(19,28)(20,29)(22,38)(23,37)(24,26)(25,36)(27,32)(33,44)(34,43)(35,39)(40,42)(41,45);(6,8)(7,10)(9,12)(11,13)(14,17,20,33)(15,22,25,40)(16,27,29,43)(18,32,19,34)(21,35,36,45)(23,39,24,41)(26,38,37,42)(28,31,30,44)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2970);