D₁₁₆₂₅ #72223926

order	23250
automorphism	Aut(G) order 69750000
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₁₁₆₂₅

Splittings

Fitting, derived, Hall	C₃×C₃₁×C₁₂₅ ⋊ C₂
Hall	C₃₁×C₁₂₅ ⋊ D₃
Hall	C₃×C₁₂₅ ⋊ D₃₁
Sylow	C₁₂₅ ⋊ D₉₃
Hall	C₃×C₃₁ ⋊ D₁₂₅
Sylow	C₃₁ ⋊ D₃₇₅
Sylow	C₃ ⋊ D₃₈₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₃₁×C₁₂₅	C₂
Frattini	C₂₅	D₄₆₅
socle	C₃×C₅×C₃₁	D₂₅
LCS	G ▷ C₃×C₃₁×C₁₂₅
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 159
transitive degree	μ_t(G) = 11625

A minimal faithful permutation representation — 2 generators on 159 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)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159);(2,3)(5,128)(6,127)(7,126)(8,125)(9,124)(10,123)(11,122)(12,121)(13,120)(14,119)(15,118)(16,117)(17,116)(18,115)(19,114)(20,113)(21,112)(22,111)(23,110)(24,109)(25,108)(26,107)(27,106)(28,105)(29,104)(30,103)(31,102)(32,101)(33,100)(34,99)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)

Polycyclic presentation:

2,3,5,31,5^2|c^5=e,e^5=f,b^a=b^2,c^a=c^4e^4f^4,d^a=d^30,e^a=e^4f^4,f^a=f^4

Conjugacy classes & character table

5814 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₁₆₂₅
maximal normal subgroup of	C₂×D₁₁₆₂₅
maximal quotient of	Dic₁₁₆₂₅, C₂×D₁₁₆₂₅

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, 31, 5, 5]);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^4*F.5^4*F.6^4);;
gap> SetConjugate(coll, 4, 1, F.4^30);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.6^4);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11625 #72223926