C₅×#13078 #11935798

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13078

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1893
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

454 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 146 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)(6,7,11,18,13,17,12,36)(8,19,23,49,22,46,34,50)(9,24,26,45,27,15,35,44)(10,28,31,38,33,60,32,16)(14,37,40,21,42,56,41,29)(20,48,53,68,52,65,54,69)(25,58,59,47,30,61,62,43)(39,64,66,55,57,63,67,51);(6,8,10,29,12,23,32,37,13,22,33,21,11,34,31,56)(7,14,38,19,18,41,60,50,17,42,16,46,36,40,28,49)(9,20,30,51,26,54,62,63,27,52,25,55,35,53,59,64)(15,39,43,48,45,66,61,68,24,57,47,65,44,67,58,69);(6,9)(7,15)(8,20)(10,30)(11,35)(12,26)(13,27)(14,39)(16,47)(17,24)(18,45)(19,48)(21,55)(22,52)(23,54)(25,33)(28,58)(29,51)(31,59)(32,62)(34,53)(36,44)(37,63)(38,43)(40,67)(41,66)(42,57)(46,65)(49,69)(50,68)(56,64)(60,61)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6066);