D₃₀₅₅₅ #72295840

order	61110
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₀₅₅₅

Splittings

Fitting, derived, Hall	C₅×C₇×C₉×C₉₇ ⋊ C₂
Hall	C₇×C₉×C₉₇ ⋊ D₅
Hall	C₅×C₉×C₉₇ ⋊ D₇
Hall	C₅×C₇×C₉₇ ⋊ D₉
Hall	C₉×C₉₇ ⋊ D₃₅
Hall	C₇×C₉₇ ⋊ D₄₅
Hall	C₅×C₉₇ ⋊ D₆₃
Hall	C₅×C₇×C₉ ⋊ D₉₇
Sylow	C₉₇ ⋊ D₃₁₅
Hall	C₇×C₉ ⋊ D₄₈₅
Hall	C₅×C₉ ⋊ D₆₇₉
Hall	C₅×C₇ ⋊ D₈₇₃
Sylow	C₉ ⋊ D₃₃₉₅
Sylow	C₇ ⋊ D₄₃₆₅
Sylow	C₅ ⋊ D₆₁₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₇×C₉×C₉₇	C₂
Frattini	C₃	D₁₀₁₈₅
socle	C₃×C₅×C₇×C₉₇	D₃
LCS	G ▷ C₅×C₇×C₉×C₉₇
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 118
transitive degree	μ_t(G) = 30555

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

Polycyclic presentation:

2,3,5,7,97,3|b^3=f,b^a=b^2f^2,c^a=c^4,d^a=d^6,e^a=e^96,f^a=f^2

Conjugacy classes & character table

15279 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, 97, 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.5^96);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30555 #72295840