Dic₁₃₈₉₉ #72290220

order	55596
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₈₉₉

Splittings

derived, Hall	C₃×C₄₁×C₁₁₃ ⋊ C₄
Hall	C₄₁×C₁₁₃ ⋊ Dic₃
Hall	C₃×C₁₁₃ ⋊ Dic₄₁
Hall	C₃×C₄₁ ⋊ Dic₁₁₃
Sylow	C₁₁₃ ⋊ Dic₁₂₃
Sylow	C₄₁ ⋊ Dic₃₃₉
Sylow	C₃ ⋊ Dic₄₆₃₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₈₉₉
commutator, hypocenter	C₃×C₄₁×C₁₁₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₄₁×C₁₁₃	C₂
LCS	G ▷ C₃×C₄₁×C₁₁₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 161
transitive degree	μ_t(G) = 55596

A minimal faithful permutation representation — 2 generators on 161 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);(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)(50,161)(51,160)(52,159)(53,158)(54,157)(55,156)(56,155)(57,154)(58,153)(59,152)(60,151)(61,150)(62,149)(63,148)(64,147)(65,146)(66,145)(67,144)(68,143)(69,142)(70,141)(71,140)(72,139)(73,138)(74,137)(75,136)(76,135)(77,134)(78,133)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)(85,126)(86,125)(87,124)(88,123)(89,122)(90,121)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)

Polycyclic presentation:

2^2,3,41,113|a^2=b,c^a=c^2,d^a=d^40,e^a=e^112

Conjugacy classes & character table

13902 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, 113]);;
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.5^112);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13899 #72290220