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D20160 #72274337

order40320
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD20160

Splittings

FittingC5×C7×C9×C64 ⋊ C2
HallC5×C7×C9 ⋊ D64
HallC7×C9 ⋊ D320
HallC5×C9 ⋊ D448
HallC5×C7 ⋊ D576
SylowC9 ⋊ D2240
SylowC7 ⋊ D2880
SylowC5 ⋊ D4032

Subgroups

subgroupG/H
centerC2D10080
commutator, cosocleC5×C7×C9×C32C22
FrattiniC3×C32C2×D105
FittingC5×C7×C9×C64C2
socleC2×C3×C5×C7D96
hypercenterC64D315
hypocenterC5×C7×C9D64
LCSG ▷ C5×C7×C9×C32 ▷ C5×C7×C9×C16 ▷ C5×C7×C8×C9 ▷ C4×C5×C7×C9 ▷ C2×C5×C7×C9 ▷ C5×C7×C9
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64

240 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 85
transitive degreeμt(G) = 20160

A minimal faithful permutation representation — 2 generators on 85 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73)(74,75,76,77,78)(79,80,81,82,83,84,85);(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(66,73)(67,72)(68,71)(69,70)(75,78)(76,77)(80,85)(81,84)(82,83)

Polycyclic presentation:

2^7,3,5,7,3|b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=g,h^3=k,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^2k^2,i^a=i^4,j^a=j^6,k^a=k^2

Conjugacy classes & character table

10083 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 5, 7, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.11^2);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.10^6);;
gap> SetConjugate(coll, 11, 1, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);