D₂₆₀₁₉ #72289134

order	52038
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₆₀₁₉

Splittings

Fitting, derived, Hall	C₉×C₄₉×C₅₉ ⋊ C₂
Hall	C₄₉×C₅₉ ⋊ D₉
Hall	C₉×C₅₉ ⋊ D₄₉
Hall	C₉×C₄₉ ⋊ D₅₉
Sylow	C₅₉ ⋊ D₄₄₁
Sylow	C₄₉ ⋊ D₅₃₁
Sylow	C₉ ⋊ D₂₈₉₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₄₉×C₅₉	C₂
Frattini	C₃×C₇	D₁₂₃₉
socle	C₃×C₇×C₅₉	D₂₁
LCS	G ▷ C₉×C₄₉×C₅₉
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 26019

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

Polycyclic presentation:

2,3,7,59,3,7|b^3=e,c^7=f,b^a=b^2e^2,c^a=c^6f^6,d^a=d^58,e^a=e^2,f^a=f^6

Conjugacy classes & character table

13011 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");;
gap> coll := SingleCollector(F, [2, 3, 7, 59, 3, 7]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2);;
gap> SetConjugate(coll, 3, 1, F.3^6*F.6^6);;
gap> SetConjugate(coll, 4, 1, F.4^58);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26019 #72289134