C₅×#13698 #11936418

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13698

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

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

normal subgroups only

This group has 190 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) = 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)(12,22,24,45)(13,28,31,60)(14,32,34,62)(15,35,33,20)(17,36,37,25)(18,40,42,23)(19,43,44,29)(26,56,52,73)(27,57,53,48)(30,61,50,59)(38,68,67,72)(39,55,63,66)(41,51,58,69)(46,65,70,49)(47,64,71,54);(10,12,23,49,14,26,50,66,33,53,69,68,43,64,31,36,16,24,40,65,34,52,30,55,15,27,51,72,29,54,13,25)(11,17,28,47,19,38,58,57,35,63,61,73,62,70,42,45,21,37,60,71,44,67,41,48,20,39,59,56,32,46,18,22);(7,9)(12,24)(17,37)(22,45)(25,36)(26,52)(27,53)(38,67)(39,63)(46,70)(47,71)(48,57)(49,65)(54,64)(55,66)(56,73)(68,72)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6686);