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D22680 #72277495

order45360
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD22680

Splittings

FittingC5×C7×C8×C81 ⋊ C2
HallC5×C7×C81 ⋊ D8
HallC7×C81 ⋊ D40
HallC5×C81 ⋊ D56
SylowC81 ⋊ D280
HallC5×C7 ⋊ D648
SylowC7 ⋊ D3240
SylowC5 ⋊ D4536

Subgroups

subgroupG/H
centerC2D11340
commutator, cosocleC4×C5×C7×C81C22
FrattiniC4×C27C2×D105
FittingC5×C7×C8×C81C2
socleC2×C3×C5×C7D108
hypercenterC8D2835
hypocenterC5×C7×C81D8
LCSG ▷ C4×C5×C7×C81 ▷ C2×C5×C7×C81 ▷ C5×C7×C81
UCSC1 ◁ C2 ◁ C4 ◁ C8

220 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 83 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) = 101
transitive degreeμt(G) = 22680

A minimal faithful permutation representation — 2 generators on 101 points:

(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89)(90,91,92,93,94)(95,96,97,98,99,100,101);(2,8)(3,7)(4,6)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(91,94)(92,93)(96,101)(97,100)(98,99)

Polycyclic presentation:

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

Conjugacy classes & character table

11343 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 7, 3, 3, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.8^2*F.9^2*F.10^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2*F.10^2);;
gap> SetConjugate(coll, 9, 1, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);