Dic₁₄₇₂₉ #72291695

order	58916
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₇₂₉

Splittings

derived, Hall	C₁₁×C₁₃×C₁₀₃ ⋊ C₄
Hall	C₁₃×C₁₀₃ ⋊ Dic₁₁
Hall	C₁₁×C₁₀₃ ⋊ Dic₁₃
Hall	C₁₁×C₁₃ ⋊ Dic₁₀₃
Sylow	C₁₀₃ ⋊ Dic₁₄₃
Sylow	C₁₃ ⋊ Dic₁₁₃₃
Sylow	C₁₁ ⋊ Dic₁₃₃₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₇₂₉
commutator, hypocenter	C₁₁×C₁₃×C₁₀₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₃×C₁₀₃	C₂
LCS	G ▷ C₁₁×C₁₃×C₁₀₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 131
transitive degree	μ_t(G) = 58916

A minimal faithful permutation representation — 2 generators on 131 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(30,131)(31,130)(32,129)(33,128)(34,127)(35,126)(36,125)(37,124)(38,123)(39,122)(40,121)(41,120)(42,119)(43,118)(44,117)(45,116)(46,115)(47,114)(48,113)(49,112)(50,111)(51,110)(52,109)(53,108)(54,107)(55,106)(56,105)(57,104)(58,103)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)

Polycyclic presentation:

2^2,11,13,103|a^2=b,c^a=c^10,d^a=d^12,e^a=e^102

Conjugacy classes & character table

14732 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, 11, 13, 103]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^10);;
gap> SetConjugate(coll, 4, 1, F.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^102);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14729 #72291695