Dic₁₃₄₀₇ #72289218

order	53628
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₄₀₇

Splittings

derived, Hall	C₃×C₄₁×C₁₀₉ ⋊ C₄
Hall	C₄₁×C₁₀₉ ⋊ Dic₃
Hall	C₃×C₁₀₉ ⋊ Dic₄₁
Hall	C₃×C₄₁ ⋊ Dic₁₀₉
Sylow	C₁₀₉ ⋊ Dic₁₂₃
Sylow	C₄₁ ⋊ Dic₃₂₇
Sylow	C₃ ⋊ Dic₄₄₆₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₄₀₇
commutator, hypocenter	C₃×C₄₁×C₁₀₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₄₁×C₁₀₉	C₂
LCS	G ▷ C₃×C₄₁×C₁₀₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 157
transitive degree	μ_t(G) = 53628

A minimal faithful permutation representation — 2 generators on 157 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);(1,3,2,4)(6,7)(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)(50,157)(51,156)(52,155)(53,154)(54,153)(55,152)(56,151)(57,150)(58,149)(59,148)(60,147)(61,146)(62,145)(63,144)(64,143)(65,142)(66,141)(67,140)(68,139)(69,138)(70,137)(71,136)(72,135)(73,134)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)

Polycyclic presentation:

2^2,3,41,109|a^2=b,c^a=c^2,d^a=d^40,e^a=e^108

Conjugacy classes & character table

13410 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, 3, 41, 109]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^40);;
gap> SetConjugate(coll, 5, 1, F.5^108);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13407 #72289218