C₅×#10009 #11932729

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10004
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10009

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 528 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) = 37
transitive degree	μ_t(G) = 320

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

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

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, 2997);