D₃₂₄₄₅ #72302145

order	64890
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₂₄₄₅

Splittings

Fitting, derived, Hall	C₅×C₇×C₉×C₁₀₃ ⋊ C₂
Hall	C₇×C₉×C₁₀₃ ⋊ D₅
Hall	C₅×C₉×C₁₀₃ ⋊ D₇
Hall	C₅×C₇×C₁₀₃ ⋊ D₉
Hall	C₉×C₁₀₃ ⋊ D₃₅
Hall	C₇×C₁₀₃ ⋊ D₄₅
Hall	C₅×C₁₀₃ ⋊ D₆₃
Hall	C₅×C₇×C₉ ⋊ D₁₀₃
Sylow	C₁₀₃ ⋊ D₃₁₅
Hall	C₇×C₉ ⋊ D₅₁₅
Hall	C₅×C₉ ⋊ D₇₂₁
Hall	C₅×C₇ ⋊ D₉₂₇
Sylow	C₉ ⋊ D₃₆₀₅
Sylow	C₇ ⋊ D₄₆₃₅
Sylow	C₅ ⋊ D₆₄₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₇×C₉×C₁₀₃	C₂
Frattini	C₃	D₁₀₈₁₅
socle	C₃×C₅×C₇×C₁₀₃	D₃
LCS	G ▷ C₅×C₇×C₉×C₁₀₃
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 124
transitive degree	μ_t(G) = 32445

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

Polycyclic presentation:

2,3,5,7,103,3|b^3=f,b^a=b^2f^2,c^a=c^4,d^a=d^6,e^a=e^102,f^a=f^2

Conjugacy classes & character table

16224 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, 103, 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.5^102);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32445 #72302145