C₅×#10115 #11932835

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10104
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10115

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

466 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 106 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)(8,15)(9,11)(10,20)(12,14)(13,16)(17,21)(18,19)(22,23,26,29)(24,30,31,37)(25,33,34,28)(27,35,36,32);(6,8,11,17)(7,13,9,19)(10,18,12,16)(14,21,20,15)(25,34)(28,33)(30,37)(32,35);(6,9,11,7)(8,15,17,21)(10,20,12,14)(13,16,19,18)(22,24)(23,27)(25,35)(26,31)(28,37)(29,36)(30,33)(32,34)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3103);