D₁₁₃₂₈ #72215959

order	22656
automorphism	Aut(G) order 42049536
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₃₂₈

Splittings

Fitting	C₃×C₅₉×C₆₄ ⋊ C₂
Hall	C₃×C₅₉ ⋊ D₆₄
Sylow	C₅₉ ⋊ D₁₉₂
Sylow	C₃ ⋊ D₃₇₇₆

Subgroups

	subgroup	G/H
center	C₂	D₅₆₆₄
commutator, cosocle	C₃×C₃₂×C₅₉	C₂²
Frattini	C₃₂	C₂×D₁₇₇
Fitting	C₃×C₅₉×C₆₄	C₂
socle	C₂×C₃×C₅₉	D₃₂
hypercenter	C₆₄	D₁₇₇
hypocenter	C₃×C₅₉	D₆₄
LCS	G ▷ C₃×C₃₂×C₅₉ ▷ C₃×C₁₆×C₅₉ ▷ C₃×C₈×C₅₉ ▷ C₃×C₄×C₅₉ ▷ C₂×C₃×C₅₉ ▷ C₃×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 126
transitive degree	μ_t(G) = 11328

A minimal faithful permutation representation — 2 generators on 126 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67)(68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126);(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(66,67)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(81,114)(82,113)(83,112)(84,111)(85,110)(86,109)(87,108)(88,107)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(96,99)(97,98)

Polycyclic presentation:

2^7,3,59|b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^2,i^a=i^58

Conjugacy classes & character table

5667 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₆₅₆
maximal normal subgroup of	D₂₂₆₅₆
maximal quotient of	Dic₁₁₃₂₈, D₂₂₆₅₆

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 59]);;
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*F.7);;
gap> SetPower(coll, 6, F.7);;
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, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11328 #72215959