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D27132 #72290459

order54264
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD27132

Splittings

FittingC3×C4×C7×C17×C19 ⋊ C2
HallC3×C7×C17×C19 ⋊ D4
HallC7×C17×C19 ⋊ D12
HallC3×C17×C19 ⋊ D28
HallC3×C7×C19 ⋊ D68
HallC3×C7×C17 ⋊ D76
HallC17×C19 ⋊ D84
HallC7×C19 ⋊ D204
HallC7×C17 ⋊ D228
HallC3×C19 ⋊ D476
HallC3×C17 ⋊ D532
HallC3×C7 ⋊ D1292
SylowC19 ⋊ D1428
SylowC17 ⋊ D1596
SylowC7 ⋊ D3876
SylowC3 ⋊ D9044

Subgroups

subgroupG/H
center, FrattiniC2C2×D6783
commutator, socle, cosocleC2×C3×C7×C17×C19C22
FittingC3×C4×C7×C17×C19C2
hypercenterC4D6783
hypocenterC3×C7×C17×C19D4
LCSG ▷ C2×C3×C7×C17×C19 ▷ C3×C7×C17×C19
UCSC1 ◁ C2 ◁ C4

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 50
transitive degreeμt(G) = 27132

A minimal faithful permutation representation — 2 generators on 50 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);(2,4)(6,7)(9,14)(10,13)(11,12)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)

Polycyclic presentation:

2^3,3,7,17,19|b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^16,g^a=g^18

Conjugacy classes & character table

13569 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, 2, 2, 3, 7, 17, 19]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);