D₃₀₄₀₀ #72295705

order	60800
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₄₀₀

Splittings

Fitting	C₁₉×C₂₅×C₆₄ ⋊ C₂
Hall	C₁₉×C₂₅ ⋊ D₆₄
Sylow	C₂₅ ⋊ D₁₂₁₆
Sylow	C₁₉ ⋊ D₁₆₀₀

Subgroups

	subgroup	G/H
center	C₂	D₁₅₂₀₀
commutator, cosocle	C₁₉×C₂₅×C₃₂	C₂²
Frattini	C₅×C₃₂	C₂×D₉₅
Fitting	C₁₉×C₂₅×C₆₄	C₂
socle	C₂×C₅×C₁₉	D₁₆₀
hypercenter	C₆₄	D₄₇₅
hypocenter	C₁₉×C₂₅	D₆₄
LCS	G ▷ C₁₉×C₂₅×C₃₂ ▷ C₁₆×C₁₉×C₂₅ ▷ C₈×C₁₉×C₂₅ ▷ C₄×C₁₉×C₂₅ ▷ C₂×C₁₉×C₂₅ ▷ C₁₉×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 108
transitive degree	μ_t(G) = 30400

A minimal faithful permutation representation — 2 generators on 108 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);(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)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)

Polycyclic presentation:

2^7,5,19,5|b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=g,h^5=j,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^4j^4,i^a=i^18,j^a=j^4

Conjugacy classes & character table

15203 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, 2, 5, 19, 5]);;
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> SetPower(coll, 8, 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, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^4*F.10^4);;
gap> SetConjugate(coll, 9, 1, F.9^18);;
gap> SetConjugate(coll, 10, 1, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30400 #72295705