C₅×#9336 #11932056

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9336

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

360 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 69
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,7,11,15,12,17,10,16)(8,18,21,28,22,31,20,13)(9,23,25,14,26,29,24,30)(19,27,34,36,35,37,33,32)(38,39)(40,49,52,60)(41,53)(42,47)(43,57)(44,48)(45,51,61,56)(46,54)(50,65,68,59)(55,62)(58,63)(64,66,69,67);(6,8,12,22)(7,13)(9,19,26,35)(10,20,11,21)(14,27)(15,18)(16,31)(17,28)(23,32)(24,33,25,34)(29,36)(30,37)(38,40,42,56,44,52,43,51)(39,45,47,60,48,61,57,49)(41,50,54,67,55,68,58,66)(46,59,62,69,63,65,53,64);(6,9,12,26)(7,14,17,30)(8,19,22,35)(10,25,11,24)(13,27,28,37)(15,23,16,29)(18,32,31,36)(20,34,21,33)(38,41)(39,46)(40,50)(42,54)(43,58)(44,55)(45,59)(47,62)(48,63)(49,64)(51,66)(52,68)(53,57)(56,67)(60,69)(61,65)

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, 2324);