C₅×#10210 #11932930

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

Aliases

direct product

C₅×#10210

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 440 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) = 53
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(8,15)(9,10)(12,16)(13,19)(17,18)(20,21)(22,23)(24,34)(25,39)(26,32)(27,31)(28,33)(29,36)(30,40)(35,46)(37,45)(38,44)(41,48)(42,47)(43,51)(49,50)(52,53);(6,7)(8,16)(9,13)(10,19)(11,14)(12,18)(15,20)(17,21)(22,24,28,38)(23,29,33,45)(25,35,41,51)(26,37,27,36)(30,46,47,43)(31,44,32,34)(39,49,48,53)(40,52,42,50);(6,8)(7,12)(9,18)(10,15)(11,17)(13,21)(14,20)(16,19)(22,25,28,41)(23,30,33,47)(24,35,38,51)(26,40,27,42)(29,43,45,46)(31,39,32,48)(34,49,44,53)(36,50,37,52)

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, 3198);