D₂₆₁₃₂ #72289367

order	52264
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₁₃₂

Splittings

Fitting	C₄×C₄₇×C₁₃₉ ⋊ C₂
Hall	C₄₇×C₁₃₉ ⋊ D₄
Sylow	C₁₃₉ ⋊ D₁₈₈
Sylow	C₄₇ ⋊ D₅₅₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₅₃₃
commutator, socle, cosocle	C₂×C₄₇×C₁₃₉	C₂²
Fitting	C₄×C₄₇×C₁₃₉	C₂
hypercenter	C₄	D₆₅₃₃
hypocenter	C₄₇×C₁₃₉	D₄
LCS	G ▷ C₂×C₄₇×C₁₃₉ ▷ C₄₇×C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 190
transitive degree	μ_t(G) = 26132

A minimal faithful permutation representation — 2 generators on 190 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)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190);(2,4)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(53,190)(54,189)(55,188)(56,187)(57,186)(58,185)(59,184)(60,183)(61,182)(62,181)(63,180)(64,179)(65,178)(66,177)(67,176)(68,175)(69,174)(70,173)(71,172)(72,171)(73,170)(74,169)(75,168)(76,167)(77,166)(78,165)(79,164)(80,163)(81,162)(82,161)(83,160)(84,159)(85,158)(86,157)(87,156)(88,155)(89,154)(90,153)(91,152)(92,151)(93,150)(94,149)(95,148)(96,147)(97,146)(98,145)(99,144)(100,143)(101,142)(102,141)(103,140)(104,139)(105,138)(106,137)(107,136)(108,135)(109,134)(110,133)(111,132)(112,131)(113,130)(114,129)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(121,122)

Polycyclic presentation:

2^3,47,139|b^2=c,b^a=bc,d^a=d^46,e^a=e^138

Conjugacy classes & character table

13069 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, 2, 2, 47, 139]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^46);;
gap> SetConjugate(coll, 5, 1, F.5^138);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26132 #72289367