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D8580 #72213482

order17160
automorphismAut(G) order 16473600
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD8580

Splittings

FittingC3×C4×C5×C11×C13 ⋊ C2
HallC3×C5×C11×C13 ⋊ D4
HallC5×C11×C13 ⋊ D12
HallC3×C11×C13 ⋊ D20
HallC3×C5×C13 ⋊ D44
HallC3×C5×C11 ⋊ D52
HallC11×C13 ⋊ D60
HallC5×C13 ⋊ D132
HallC5×C11 ⋊ D156
HallC3×C13 ⋊ D220
HallC3×C11 ⋊ D260
HallC3×C5 ⋊ D572
SylowC13 ⋊ D660
SylowC11 ⋊ D780
SylowC5 ⋊ D1716
SylowC3 ⋊ D2860

Subgroups

subgroupG/H
center, FrattiniC2C2×D2145
commutator, socle, cosocleC2×C3×C5×C11×C13C22
FittingC3×C4×C5×C11×C13C2
hypercenterC4D2145
hypocenterC3×C5×C11×C13D4
LCSG ▷ C2×C3×C5×C11×C13 ▷ C3×C5×C11×C13
UCSC1 ◁ C2 ◁ C4

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 8580

A minimal faithful permutation representation — 2 generators on 36 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);(2,4)(6,7)(9,12)(10,11)(14,23)(15,22)(16,21)(17,20)(18,19)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

4293 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD17160, D25740
maximal normal subgroup ofD17160
maximal quotient ofDic8580, D17160, D25740

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, 11, 13]);;
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.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);