D₃₁₅₅₂ #72301488

order	63104
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₁₅₅₂

Splittings

Fitting	C₁₇×C₂₉×C₆₄ ⋊ C₂
Hall	C₁₇×C₂₉ ⋊ D₆₄
Sylow	C₂₉ ⋊ D₁₀₈₈
Sylow	C₁₇ ⋊ D₁₈₅₆

Subgroups

	subgroup	G/H
center	C₂	D₁₅₇₇₆
commutator, cosocle	C₁₇×C₂₉×C₃₂	C₂²
Frattini	C₃₂	C₂×D₄₉₃
Fitting	C₁₇×C₂₉×C₆₄	C₂
socle	C₂×C₁₇×C₂₉	D₃₂
hypercenter	C₆₄	D₄₉₃
hypocenter	C₁₇×C₂₉	D₆₄
LCS	G ▷ C₁₇×C₂₉×C₃₂ ▷ C₁₆×C₁₇×C₂₉ ▷ C₈×C₁₇×C₂₉ ▷ C₄×C₁₇×C₂₉ ▷ C₂×C₁₇×C₂₉ ▷ C₁₇×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 110
transitive degree	μ_t(G) = 31552

A minimal faithful permutation representation — 2 generators on 110 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);(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)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(83,110)(84,109)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)

Polycyclic presentation:

2^7,17,29|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^16,i^a=i^28

Conjugacy classes & character table

15779 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 17, 29]);;
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.8^16);;
gap> SetConjugate(coll, 9, 1, F.9^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31552 #72301488