Dic₁₄₈₄₇ #72292142

order	59388
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₈₄₇

Splittings

derived, Hall	C₃×C₄₉×C₁₀₁ ⋊ C₄
Hall	C₄₉×C₁₀₁ ⋊ Dic₃
Hall	C₃×C₁₀₁ ⋊ Dic₄₉
Hall	C₃×C₄₉ ⋊ Dic₁₀₁
Sylow	C₁₀₁ ⋊ Dic₁₄₇
Sylow	C₄₉ ⋊ Dic₃₀₃
Sylow	C₃ ⋊ Dic₄₉₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₄₈₄₇
commutator, hypocenter	C₃×C₄₉×C₁₀₁	C₄
Frattini	C₂×C₇	D₂₁₂₁
Fitting, cosocle	C₂×C₃×C₄₉×C₁₀₁	C₂
socle, prime radical	C₂×C₃×C₇×C₁₀₁	D₇
LCS	G ▷ C₃×C₄₉×C₁₀₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 157
transitive degree	μ_t(G) = 59388

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)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(58,157)(59,156)(60,155)(61,154)(62,153)(63,152)(64,151)(65,150)(66,149)(67,148)(68,147)(69,146)(70,145)(71,144)(72,143)(73,142)(74,141)(75,140)(76,139)(77,138)(78,137)(79,136)(80,135)(81,134)(82,133)(83,132)(84,131)(85,130)(86,129)(87,128)(88,127)(89,126)(90,125)(91,124)(92,123)(93,122)(94,121)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)

Polycyclic presentation:

2^2,3,7,101,7|a^2=b,d^7=f,c^a=c^2,d^a=d^6f^6,e^a=e^100,f^a=f^6

Conjugacy classes & character table

14850 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", "f");;
gap> coll := SingleCollector(F, [2, 2, 3, 7, 101, 7]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^6*F.6^6);;
gap> SetConjugate(coll, 5, 1, F.5^100);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14847 #72292142