C₅×#9440 #11932160

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9431
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9440

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

434 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 118 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)(6,7,10,14,11,15,9,12)(8,16,18,13,19,20,17,21)(22,23)(24,32,25,28)(26,31)(27,30)(29,34,36,33)(35,37);(7,12,15,14)(13,16,21,20)(22,24,26,25)(23,28)(27,34,35,33)(29,30)(31,32)(36,37);(6,8)(7,13)(9,17)(10,18)(11,19)(12,16)(14,20)(15,21)(22,25,26,24)(23,29,31,36)(27,33,35,34)(28,30,32,37)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2428);