D₂₆₂₅₇ #72289618

order	52514
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₆₂₅₇

Splittings

Fitting, derived, Hall	C₇×C₃₁×C₁₂₁ ⋊ C₂
Hall	C₃₁×C₁₂₁ ⋊ D₇
Hall	C₇×C₁₂₁ ⋊ D₃₁
Hall	C₇×C₃₁ ⋊ D₁₂₁
Sylow	C₁₂₁ ⋊ D₂₁₇
Sylow	C₃₁ ⋊ D₈₄₇
Sylow	C₇ ⋊ D₃₇₅₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₃₁×C₁₂₁	C₂
Frattini	C₁₁	D₂₃₈₇
socle	C₇×C₁₁×C₃₁	D₁₁
LCS	G ▷ C₇×C₃₁×C₁₂₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 159
transitive degree	μ_t(G) = 26257

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);(2,7)(3,6)(4,5)(9,128)(10,127)(11,126)(12,125)(13,124)(14,123)(15,122)(16,121)(17,120)(18,119)(19,118)(20,117)(21,116)(22,115)(23,114)(24,113)(25,112)(26,111)(27,110)(28,109)(29,108)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(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:

2,7,11,31,11|c^11=e,b^a=b^6,c^a=c^10e^10,d^a=d^30,e^a=e^10

Conjugacy classes & character table

13130 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");;
gap> coll := SingleCollector(F, [2, 7, 11, 31, 11]);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^6);;
gap> SetConjugate(coll, 3, 1, F.3^10*F.5^10);;
gap> SetConjugate(coll, 4, 1, F.4^30);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26257 #72289618