C₅×#10339 #11933059

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10336
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10339

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂⁵	C₂³×C₅
prime radical	C₂⁵×C₅	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁵×C₅ ◁ G

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

normal subgroups only

This group has 738 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)(8,12,19,26)(9,13,20,28)(11,16,22,14)(17,31,32,27)(18,24,34,25)(21,30,35,29)(23,36,37,33);(6,8,10,19)(7,12,15,26)(9,17,20,32)(11,18,22,34)(13,27,28,31)(14,25,16,24)(21,33,35,36)(23,30,37,29);(6,9,11,21)(7,13,16,30)(8,17,18,33)(10,20,22,35)(12,23,24,31)(14,29,15,28)(19,32,34,36)(25,27,26,37)

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, 3327);