C₅×#11914 #11934634

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11914

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1530
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	#97	C₂³×C₅
prime radical	C₂³×C₅	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#405 ◁ G

272 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 320

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

(1,2,3,4,5)(6,7,10,16,11,17,31,47,12,18,30,48,33,15,42,44)(8,19,21,50,22,13,34,36,23,37,55,45,32,38,28,61)(9,24,26,56,27,41,59,46,25,43,29,63,57,14,39,40)(20,35,52,65,53,64,62,68,51,66,54,67,60,49,58,69);(6,8,12,23)(7,13,18,38)(9,20,25,51)(10,28,30,34)(11,32,33,22)(14,35,41,66)(15,37,17,19)(16,45,48,50)(21,31,55,42)(24,49,43,64)(26,58,29,62)(27,60,57,53)(36,44,61,47)(39,52,59,54)(40,68,46,69)(56,65,63,67);(6,9,12,25)(7,14)(8,20,23,51)(10,29,30,26)(11,27,33,57)(13,35)(15,43)(16,46)(17,24)(18,41)(19,49)(21,54,55,52)(22,53,32,60)(28,62,34,58)(31,39,42,59)(36,67)(37,64)(38,66)(40,48)(44,56)(45,69)(47,63)(50,68)(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, 4902);