C₅×#13710 #11936430

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13709
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13710

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2226
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₅×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

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

normal subgroups only

This group has 160 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)(7,14)(8,18,21,36)(9,12)(10,23)(11,17)(15,31,20,37)(16,25)(19,32,28,34)(22,33)(24,27,35,29)(26,30);(6,7,12,11,13,17,9,14)(8,15,28,27,21,20,19,29)(10,16,25,23,26,33,22,30)(18,24,32,37,36,35,34,31);(6,8)(7,15)(9,19)(10,24)(11,27)(12,28)(13,21)(14,29)(16,32)(17,20)(18,30)(22,31)(23,36)(25,37)(26,35)(33,34)

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, 6698);