C₅×#9419 #11932139

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×#9419

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

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

normal subgroups only

This group has 424 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) = 640

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

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

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, 2407);