C₅×#13695 #11936415

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13695

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₂³×C₅	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

194 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 73
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 73 points:

(1,2,3,4,5)(6,7)(8,9)(10,11,16,21)(12,22,27,48)(13,28,31,60)(14,32,34,62)(15,35,33,20)(17,36,39,24)(18,40,42,23)(19,43,44,29)(25,54,56,73)(26,57,55,47)(30,61,50,59)(37,67,68,72)(38,53,63,66)(41,51,58,69)(45,65,70,49)(46,64,71,52);(10,12,23,49,14,25,50,66,33,55,69,67,43,64,31,36,16,27,40,65,34,56,30,53,15,26,51,72,29,52,13,24)(11,17,28,46,19,37,58,57,35,63,61,73,62,70,42,48,21,39,60,71,44,68,41,47,20,38,59,54,32,45,18,22);(7,9)

Conjugacy classes & character table

350 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6683);