C₅×#10211 #11932931

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10208
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10211

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 406 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)(6,7,14,20)(8,21,27,35)(9,28,32,23)(10,33,38,56)(11,39,44,52)(12,22,46,58)(13,19,42,55)(15,47,25,29)(16,49,53,37)(17,43,57,69)(18,36,59,62)(24,60,51,68)(26,50,61,67)(30,66,45,40)(31,64,65,34)(41,63,54,48);(6,8,13,26)(7,9,19,31)(10,34,37,23)(11,40,43,24)(12,15,36,48)(14,27,42,61)(16,50,33,21)(17,54,52,29)(18,45,58,68)(20,32,55,65)(22,60,59,30)(25,62,63,46)(28,38,64,49)(35,53,67,56)(39,47,57,41)(44,66,69,51);(6,9,13,31)(7,15,19,48)(8,22,26,59)(10,35,37,67)(11,41,43,47)(12,45,36,68)(14,32,42,65)(16,51,33,66)(17,34,52,23)(18,27,58,61)(20,25,55,63)(21,49,50,38)(24,56,40,53)(28,57,64,39)(29,44,54,69)(30,62,60,46)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3199);