C₅×#13692 #11936412

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13692

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2206
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

216 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 70 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)(7,11)(8,18)(9,12)(10,22)(14,28)(15,21)(16,17)(19,30)(20,35)(23,26)(24,32)(25,31)(27,34)(29,37)(33,36);(6,7,12,16,13,17,9,11)(8,14,27,33,20,29,19,26)(10,15,24,31,25,32,21,22)(18,23,30,37,35,36,34,28);(6,8)(7,14)(9,19)(10,23)(11,26)(12,27)(13,20)(15,30)(16,33)(17,29)(18,22)(21,28)(24,37)(25,36)(31,35)(32,34)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6680);