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D28860 #72292467

order57720
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD28860

Splittings

FittingC3×C4×C5×C13×C37 ⋊ C2
HallC3×C5×C13×C37 ⋊ D4
HallC5×C13×C37 ⋊ D12
HallC3×C13×C37 ⋊ D20
HallC3×C5×C37 ⋊ D52
HallC13×C37 ⋊ D60
HallC3×C5×C13 ⋊ D148
HallC5×C37 ⋊ D156
HallC3×C37 ⋊ D260
HallC5×C13 ⋊ D444
HallC3×C13 ⋊ D740
SylowC37 ⋊ D780
HallC3×C5 ⋊ D1924
SylowC13 ⋊ D2220
SylowC5 ⋊ D5772
SylowC3 ⋊ D9620

Subgroups

subgroupG/H
center, FrattiniC2C2×D7215
commutator, socle, cosocleC2×C3×C5×C13×C37C22
FittingC3×C4×C5×C13×C37C2
hypercenterC4D7215
hypocenterC3×C5×C13×C37D4
LCSG ▷ C2×C3×C5×C13×C37 ▷ C3×C5×C13×C37
UCSC1 ◁ C2 ◁ C4

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 62
transitive degreeμt(G) = 28860

A minimal faithful permutation representation — 2 generators on 62 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);(2,4)(6,7)(9,12)(10,11)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Polycyclic presentation:

2^3,3,5,13,37|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^12,g^a=g^36

Conjugacy classes & character table

14433 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, 5, 13, 37]);;
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.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);