C₅×#13728 #11936448

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#13728

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₂×C₅×D₃₂	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

238 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 86 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)(7,9)(10,11)(12,22)(13,28)(14,18)(15,34)(16,35)(17,24)(19,23)(20,29)(21,33)(25,42)(26,55)(27,56)(30,57)(31,61)(32,62)(36,51)(37,65)(38,66)(39,47)(40,60)(41,67)(43,46)(44,50)(45,54)(48,59)(49,58)(52,63)(53,72)(64,73)(68,70)(69,71);(10,12,23,46,15,26,48,66,33,54,67,73,60,71,30,51,14,25,47,70,32,53,31,52,16,27,49,65,29,50,13,24)(11,17,28,44,20,37,58,56,35,63,61,72,62,68,39,42,18,36,57,69,40,64,41,45,21,38,59,55,34,43,19,22);(6,7,8,9)(11,18)(17,36)(19,39)(20,40)(21,35)(22,42)(28,57)(34,62)(37,64)(38,63)(41,58)(43,68)(44,69)(45,56)(55,72)(59,61)

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, 6716);