C₅×#12464 #11935184

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

Aliases

direct product

C₅×#12464

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

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, 5452);