C₅×#10648 #11933368

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#10648

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

236 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 640

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

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3636);