C₅×#11911 #11934631

order	1280
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11911

Subgroups

	subgroup	G/H
center	C₄×C₅	#329
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	#97	C₂³×C₅
socle	C₂×C₅	C₂×#329
prime radical	C₅×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

318 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 160

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

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

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, 4899);