C₅×#9984 #11932704

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

Aliases

direct product

C₅×#9984

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂×C₅×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

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

normal subgroups only

This group has 544 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) = 37
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 37 points:

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

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, 2972);