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D22176 #72275715

order44352
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD22176

Splittings

FittingC7×C9×C11×C32 ⋊ C2
HallC7×C9×C11 ⋊ D32
HallC9×C11 ⋊ D224
HallC7×C11 ⋊ D288
HallC7×C9 ⋊ D352
SylowC11 ⋊ D2016
SylowC9 ⋊ D2464
SylowC7 ⋊ D3168

Subgroups

subgroupG/H
centerC2D11088
commutator, cosocleC7×C9×C11×C16C22
FrattiniC3×C16C2×D231
FittingC7×C9×C11×C32C2
socleC2×C3×C7×C11D48
hypercenterC32D693
hypocenterC7×C9×C11D32
LCSG ▷ C7×C9×C11×C16 ▷ C7×C8×C9×C11 ▷ C4×C7×C9×C11 ▷ C2×C7×C9×C11 ▷ C7×C9×C11
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32

204 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 59
transitive degreeμt(G) = 22176

A minimal faithful permutation representation — 2 generators on 59 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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,41)(35,40)(36,39)(37,38)(43,48)(44,47)(45,46)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2^6,3,7,11,3|b^2=cd,c^2=de,d^2=ef,e^2=f,g^3=j,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2j^2,h^a=h^6,i^a=i^10,j^a=j^2

Conjugacy classes & character table

11091 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", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 7, 11, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.10^2);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 9, 1, F.9^10);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);