C₅×#11337 #11934057

order	1280
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#11337

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#127
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂²×C₅	#392
prime radical	C₂⁴×C₅	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ G

686 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 254 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(6,7,8,9)(10,11)(12,21)(13,26)(14,31)(15,19)(16,20)(17,23)(18,29)(22,32)(24,34)(25,33)(27,35)(28,37)(30,36)(38,40)(39,41);(10,12,14,23,16,25,15,24)(11,17,19,21,20,34,31,33)(13,27,28,38,30,39,29,22)(18,35,36,32,37,41,26,40);(10,13,14,29,16,30,15,28)(11,18,19,26,20,37,31,36)(12,22,24,27,25,38,23,39)(17,32,33,41,34,40,21,35)

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, 4325);