C₅×#12463 #11935183

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12463
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#12463

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1675
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ G

172 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 74 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)(6,7,13,19)(8,20,25,52)(9,26)(10,31,33,17)(11,16,35,47)(12,36,32,18)(14,37,41,22)(15,42)(21,53)(23,50,60,40)(24,39,58,51)(27,46)(28,62)(29,48)(30,43)(34,44)(38,55)(45,63)(49,64)(54,61)(56,66)(57,65)(59,68)(67,69);(6,8,11,23,12,24,33,37,13,25,35,60,32,58,10,22)(7,14,31,51,18,40,47,52,19,41,17,39,36,50,16,20)(9,21,34,59,28,55,42,64,30,57,63,69,29,56,27,54)(15,38,62,68,44,53,26,61,46,66,48,67,45,65,43,49);(6,9)(7,15)(8,21)(10,27)(11,34)(12,28)(13,30)(14,38)(16,43)(17,48)(18,44)(19,46)(20,49)(22,54)(23,59)(24,55)(25,57)(26,47)(29,32)(31,62)(33,42)(35,63)(36,45)(37,64)(39,67)(40,53)(41,66)(50,65)(51,68)(52,61)(56,58)(60,69)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5451);