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D27489 #72290790

order54978
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD27489

Splittings

Fitting, derived, HallC3×C11×C17×C49 ⋊ C2
HallC11×C17×C49 ⋊ D3
HallC3×C17×C49 ⋊ D11
HallC3×C11×C49 ⋊ D17
HallC17×C49 ⋊ D33
HallC3×C11×C17 ⋊ D49
HallC11×C49 ⋊ D51
HallC11×C17 ⋊ D147
HallC3×C49 ⋊ D187
HallC3×C17 ⋊ D539
SylowC49 ⋊ D561
HallC3×C11 ⋊ D833
SylowC17 ⋊ D1617
SylowC11 ⋊ D2499
SylowC3 ⋊ D9163

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC3×C11×C17×C49C2
FrattiniC7D3927
socleC3×C7×C11×C17D7
LCSG ▷ C3×C11×C17×C49
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 80
transitive degreeμt(G) = 27489

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

Polycyclic presentation:

2,3,7,11,17,7|c^7=f,b^a=b^2,c^a=c^6f^6,d^a=d^10,e^a=e^16,f^a=f^6

Conjugacy classes & character table

13746 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");;
gap> coll := SingleCollector(F, [2, 3, 7, 11, 17, 7]);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^6*F.6^6);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);