C₅×#9863 #11932583

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#9863

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#127
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#390
prime radical	C₅×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

412 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 142 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)(8,14)(9,17)(10,12)(11,15)(13,18)(16,20)(19,21)(22,23,26,30,27,31,25,28)(24,32,35,29,33,36,34,37);(6,8,9,15,10,16,18,21)(7,11,13,14,12,19,17,20)(23,28,31,30)(24,33)(29,36,37,32)(34,35);(7,12)(11,19)(13,17)(14,20)(22,24,25,34,27,33,26,35)(23,29,30,36,31,37,28,32)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2851);