C₅×#9450 #11932170

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9443
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9450

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

424 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 122 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)(8,15,16,19)(9,12,17,18)(10,13)(11,14)(20,21)(22,23,26,31,27,30,25,29)(24,28,32,35,33,36,34,37);(6,8,10,18,11,16,20,12)(7,9,13,19,14,17,21,15)(22,24,25,32,27,33,26,34)(23,28,31,37,30,36,29,35);(6,9)(7,12)(8,13)(10,19)(11,17)(14,18)(15,20)(16,21)(22,25,27,26)(23,29,30,31)(24,32,33,34)(28,35,36,37)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2438);