D₂₆₃₂₅ #72289680

order	52650
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₆₃₂₅

Splittings

Fitting, derived, Hall	C₁₃×C₂₅×C₈₁ ⋊ C₂
Hall	C₂₅×C₈₁ ⋊ D₁₃
Hall	C₁₃×C₈₁ ⋊ D₂₅
Hall	C₁₃×C₂₅ ⋊ D₈₁
Sylow	C₈₁ ⋊ D₃₂₅
Sylow	C₂₅ ⋊ D₁₀₅₃
Sylow	C₁₃ ⋊ D₂₀₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₃×C₂₅×C₈₁	C₂
Frattini	C₅×C₂₇	D₁₉₅
socle	C₃×C₅×C₁₃	D₁₃₅
LCS	G ▷ C₁₃×C₂₅×C₈₁
UCS	C₁

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 26325

A minimal faithful permutation representation — 2 generators on 119 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);(2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)

Polycyclic presentation:

2,3,5,13,3,5,3^2|b^3=e,c^5=f,e^3=g,g^3=h,b^a=b^2e^2g^2h^2,c^a=c^4f^4,d^a=d^12,e^a=e^2g^2h^2,f^a=f^4,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

13164 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", "g", "h");;
gap> coll := SingleCollector(F, [2, 3, 5, 13, 3, 5, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 3, 1, F.3^4*F.6^4);;
gap> SetConjugate(coll, 4, 1, F.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26325 #72289680