D₃₁₈₁₅ #72301757

order	63630
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₁₈₁₅

Splittings

Fitting, derived, Hall	C₅×C₇×C₉×C₁₀₁ ⋊ C₂
Hall	C₇×C₉×C₁₀₁ ⋊ D₅
Hall	C₅×C₉×C₁₀₁ ⋊ D₇
Hall	C₅×C₇×C₁₀₁ ⋊ D₉
Hall	C₉×C₁₀₁ ⋊ D₃₅
Hall	C₇×C₁₀₁ ⋊ D₄₅
Hall	C₅×C₁₀₁ ⋊ D₆₃
Hall	C₅×C₇×C₉ ⋊ D₁₀₁
Sylow	C₁₀₁ ⋊ D₃₁₅
Hall	C₇×C₉ ⋊ D₅₀₅
Hall	C₅×C₉ ⋊ D₇₀₇
Hall	C₅×C₇ ⋊ D₉₀₉
Sylow	C₉ ⋊ D₃₅₃₅
Sylow	C₇ ⋊ D₄₅₄₅
Sylow	C₅ ⋊ D₆₃₆₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₇×C₉×C₁₀₁	C₂
Frattini	C₃	D₁₀₆₀₅
socle	C₃×C₅×C₇×C₁₀₁	D₃
LCS	G ▷ C₅×C₇×C₉×C₁₀₁
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 122
transitive degree	μ_t(G) = 31815

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

Polycyclic presentation:

2,3,5,7,101,3|b^3=f,b^a=b^2f^2,c^a=c^4,d^a=d^6,e^a=e^100,f^a=f^2

Conjugacy classes & character table

15909 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, 5, 7, 101, 3]);;
gap> SetPower(coll, 2, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^100);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31815 #72301757