D₂₉₉₇₂ #72295319

order	59944
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₉₉₇₂

Splittings

Fitting	C₄×C₅₉×C₁₂₇ ⋊ C₂
Hall	C₅₉×C₁₂₇ ⋊ D₄
Sylow	C₁₂₇ ⋊ D₂₃₆
Sylow	C₅₉ ⋊ D₅₀₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₄₉₃
commutator, socle, cosocle	C₂×C₅₉×C₁₂₇	C₂²
Fitting	C₄×C₅₉×C₁₂₇	C₂
hypercenter	C₄	D₇₄₉₃
hypocenter	C₅₉×C₁₂₇	D₄
LCS	G ▷ C₂×C₅₉×C₁₂₇ ▷ C₅₉×C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 190
transitive degree	μ_t(G) = 29972

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

Polycyclic presentation:

2^3,59,127|b^2=c,b^a=bc,d^a=d^58,e^a=e^126

Conjugacy classes & character table

14989 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, 59, 127]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^58);;
gap> SetConjugate(coll, 5, 1, F.5^126);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D29972 #72295319