C₅×#10499 #11933219

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10496
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10499

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

326 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 37
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7,11,15)(8,9,13,19)(10,16,21,14)(12,20,18,17)(22,23)(24,27)(25,28)(26,30)(29,34)(31,35)(32,37)(33,36);(6,8)(7,12)(9,16)(10,20)(11,13)(14,19)(15,18)(17,21)(22,24,26,32)(23,27,30,37)(25,33,34,31)(28,36,29,35);(6,9,10,18,11,19,21,12)(7,13,14,20,15,8,16,17)(23,28)(25,34)(27,35)(29,30)(31,33)(36,37)

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, 3487);