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D32175 #72301949

order64350
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD32175

Splittings

Fitting, derived, HallC9×C11×C13×C25 ⋊ C2
HallC11×C13×C25 ⋊ D9
HallC9×C13×C25 ⋊ D11
HallC9×C11×C25 ⋊ D13
HallC9×C11×C13 ⋊ D25
HallC13×C25 ⋊ D99
HallC11×C25 ⋊ D117
HallC9×C25 ⋊ D143
HallC11×C13 ⋊ D225
HallC9×C13 ⋊ D275
HallC9×C11 ⋊ D325
SylowC25 ⋊ D1287
SylowC13 ⋊ D2475
SylowC11 ⋊ D2925
SylowC9 ⋊ D3575

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC9×C11×C13×C25C2
FrattiniC3×C5D2145
socleC3×C5×C11×C13D15
LCSG ▷ C9×C11×C13×C25
UCSC1

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 58
transitive degreeμt(G) = 32175

A minimal faithful permutation representation — 2 generators on 58 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);(2,9)(3,8)(4,7)(5,6)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(36,45)(37,44)(38,43)(39,42)(40,41)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2,3,5,11,13,3,5|b^3=f,c^5=g,b^a=b^2f^2,c^a=c^4g^4,d^a=d^10,e^a=e^12,f^a=f^2,g^a=g^4

Conjugacy classes & character table

16089 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");;
gap> coll := SingleCollector(F, [2, 3, 5, 11, 13, 3, 5]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^4*F.7^4);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);