Dic₁₄₉₈₆ #72292364

order	59944
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₉₈₆

Splittings

Hall	C₅₉×C₁₂₇ ⋊ Q₈
Sylow	C₁₂₇ ⋊ Dic₁₁₈
Sylow	C₅₉ ⋊ Dic₂₅₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₄₉₃
commutator, socle, prime radical, cosocle	C₂×C₅₉×C₁₂₇	C₂²
Fitting	C₄×C₅₉×C₁₂₇	C₂
hypercenter	C₄	D₇₄₉₃
hypocenter	C₅₉×C₁₂₇	Q₈
LCS	G ▷ C₂×C₅₉×C₁₂₇ ▷ C₅₉×C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 194
transitive degree	μ_t(G) = 59944

A minimal faithful permutation representation — 2 generators on 194 points:

(1,2,3,4)(5,8,7,6)(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)(68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194);(1,5,3,7)(2,6,4,8)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(81,182)(82,181)(83,180)(84,179)(85,178)(86,177)(87,176)(88,175)(89,174)(90,173)(91,172)(92,171)(93,170)(94,169)(95,168)(96,167)(97,166)(98,165)(99,164)(100,163)(101,162)(102,161)(103,160)(104,159)(105,158)(106,157)(107,156)(108,155)(109,154)(110,153)(111,152)(112,151)(113,150)(114,149)(115,148)(116,147)(117,146)(118,145)(119,144)(120,143)(121,142)(122,141)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)

Polycyclic presentation:

2^3,59,127|a^2=c,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, 1, F.3);;
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);

Dic14986 #72292364