C₅×#9430 #11932150

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9430

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 444 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)(6,7,10,13,11,15,9,14)(8,16,18,20,19,21,17,12)(22,23,28,31,27,24,26,30)(25,29,35,37,34,32,33,36);(6,8)(7,12,15,20)(9,17)(10,18)(11,19)(13,16,14,21)(22,24,27,23)(25,32,34,29)(26,30,28,31)(33,36,35,37);(7,13)(9,10)(12,16)(14,15)(17,18)(20,21)(22,25,27,34)(23,29,24,32)(26,35,28,33)(30,37,31,36)

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, 2418);