D₃₂₃₉₁ #72302111

order	64782
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₂₃₉₁

Splittings

Fitting, derived, Hall	C₉×C₅₉×C₆₁ ⋊ C₂
Hall	C₅₉×C₆₁ ⋊ D₉
Hall	C₉×C₆₁ ⋊ D₅₉
Hall	C₉×C₅₉ ⋊ D₆₁
Sylow	C₆₁ ⋊ D₅₃₁
Sylow	C₅₉ ⋊ D₅₄₉
Sylow	C₉ ⋊ D₃₅₉₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₅₉×C₆₁	C₂
Frattini	C₃	D₁₀₇₉₇
socle	C₃×C₅₉×C₆₁	D₃
LCS	G ▷ C₉×C₅₉×C₆₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 32391

A minimal faithful permutation representation — 2 generators on 129 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)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(2,9)(3,8)(4,7)(5,6)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(70,129)(71,128)(72,127)(73,126)(74,125)(75,124)(76,123)(77,122)(78,121)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)

Polycyclic presentation:

2,3,59,61,3|b^3=e,b^a=b^2e^2,c^a=c^58,d^a=d^60,e^a=e^2

Conjugacy classes & character table

16197 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");;
gap> coll := SingleCollector(F, [2, 3, 59, 61, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2);;
gap> SetConjugate(coll, 3, 1, F.3^58);;
gap> SetConjugate(coll, 4, 1, F.4^60);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32391 #72302111