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D11220 #72223883

order22440
automorphismAut(G) order 28723200
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD11220

Splittings

FittingC3×C4×C5×C11×C17 ⋊ C2
HallC3×C5×C11×C17 ⋊ D4
HallC5×C11×C17 ⋊ D12
HallC3×C11×C17 ⋊ D20
HallC3×C5×C17 ⋊ D44
HallC11×C17 ⋊ D60
HallC3×C5×C11 ⋊ D68
HallC5×C17 ⋊ D132
HallC5×C11 ⋊ D204
HallC3×C17 ⋊ D220
HallC3×C11 ⋊ D340
SylowC17 ⋊ D660
HallC3×C5 ⋊ D748
SylowC11 ⋊ D1020
SylowC5 ⋊ D2244
SylowC3 ⋊ D3740

Subgroups

subgroupG/H
center, FrattiniC2C2×D2805
commutator, socle, cosocleC2×C3×C5×C11×C17C22
FittingC3×C4×C5×C11×C17C2
hypercenterC4D2805
hypocenterC3×C5×C11×C17D4
LCSG ▷ C2×C3×C5×C11×C17 ▷ C3×C5×C11×C17
UCSC1 ◁ C2 ◁ C4

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 11220

A minimal faithful permutation representation — 2 generators on 40 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);(2,4)(6,7)(9,12)(10,11)(14,23)(15,22)(16,21)(17,20)(18,19)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)

Polycyclic presentation:

2^3,3,5,11,17|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^10,g^a=g^16

Conjugacy classes & character table

5613 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD22440
maximal normal subgroup ofD22440
maximal quotient ofDic11220, D22440

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, 17]);;
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.7^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);