Dic₁₃₂₉₈ #72289012

order	53192
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₂₉₈

Splittings

Hall	C₆₁×C₁₀₉ ⋊ Q₈
Sylow	C₁₀₉ ⋊ Dic₁₂₂
Sylow	C₆₁ ⋊ Dic₂₁₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₆₄₉
commutator, socle, prime radical, cosocle	C₂×C₆₁×C₁₀₉	C₂²
Fitting	C₄×C₆₁×C₁₀₉	C₂
hypercenter	C₄	D₆₆₄₉
hypocenter	C₆₁×C₁₀₉	Q₈
LCS	G ▷ C₂×C₆₁×C₁₀₉ ▷ C₆₁×C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 178
transitive degree	μ_t(G) = 53192

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

Polycyclic presentation:

2^3,61,109|a^2=c,b^2=c,b^a=bc,d^a=d^60,e^a=e^108

Conjugacy classes & character table

13301 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, 61, 109]);;
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.4^60);;
gap> SetConjugate(coll, 5, 1, F.5^108);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13298 #72289012