Dic₁₃₈₀₃ #72290056

order	55212
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₈₀₃

Splittings

derived, Hall	C₃×C₄₃×C₁₀₇ ⋊ C₄
Hall	C₄₃×C₁₀₇ ⋊ Dic₃
Hall	C₃×C₁₀₇ ⋊ Dic₄₃
Hall	C₃×C₄₃ ⋊ Dic₁₀₇
Sylow	C₁₀₇ ⋊ Dic₁₂₉
Sylow	C₄₃ ⋊ Dic₃₂₁
Sylow	C₃ ⋊ Dic₄₆₀₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₈₀₃
commutator, hypocenter	C₃×C₄₃×C₁₀₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₄₃×C₁₀₇	C₂
LCS	G ▷ C₃×C₄₃×C₁₀₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 157
transitive degree	μ_t(G) = 55212

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)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(52,157)(53,156)(54,155)(55,154)(56,153)(57,152)(58,151)(59,150)(60,149)(61,148)(62,147)(63,146)(64,145)(65,144)(66,143)(67,142)(68,141)(69,140)(70,139)(71,138)(72,137)(73,136)(74,135)(75,134)(76,133)(77,132)(78,131)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,120)(90,119)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)

Polycyclic presentation:

2^2,3,43,107|a^2=b,c^a=c^2,d^a=d^42,e^a=e^106

Conjugacy classes & character table

13806 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, 43, 107]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^42);;
gap> SetConjugate(coll, 5, 1, F.5^106);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13803 #72290056