D₅₁₂₅ #72211596

order	10250
automorphism	Aut(G) order 20500000
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₅₁₂₅

Splittings

Fitting, derived, Hall	C₄₁×C₁₂₅ ⋊ C₂
Sylow	C₁₂₅ ⋊ D₄₁
Sylow	C₄₁ ⋊ D₁₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄₁×C₁₂₅	C₂
Frattini	C₂₅	D₂₀₅
socle	C₅×C₄₁	D₂₅
LCS	G ▷ C₄₁×C₁₂₅
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 166
transitive degree	μ_t(G) = 5125

A minimal faithful permutation representation — 2 generators on 166 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)(126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166);(2,125)(3,124)(4,123)(5,122)(6,121)(7,120)(8,119)(9,118)(10,117)(11,116)(12,115)(13,114)(14,113)(15,112)(16,111)(17,110)(18,109)(19,108)(20,107)(21,106)(22,105)(23,104)(24,103)(25,102)(26,101)(27,100)(28,99)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(127,166)(128,165)(129,164)(130,163)(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)

Polycyclic presentation:

2,5,41,5^2|b^5=d,d^5=e,b^a=b^4d^4e^4,c^a=c^40,d^a=d^4e^4,e^a=e^4

Conjugacy classes & character table

2564 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₅₁₂₅, D₁₅₃₇₅, D₂₅₆₂₅
maximal normal subgroup of	C₂×D₅₁₂₅
maximal quotient of	C₂×D₅₁₂₅, Dic₅₁₂₅, D₁₅₃₇₅, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 5, 41, 5, 5]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^4*F.4^4*F.5^4);;
gap> SetConjugate(coll, 3, 1, F.3^40);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.5^4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D5125 #72211596