C₅×#13667 #11936387

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13667
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13667

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×M₃₂ ◁ G

294 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 160

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

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

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, 6655);