Dic₁₄₂₄₁ #72290770

order	56964
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₂₄₁

Splittings

derived, Hall	C₃×C₄₇×C₁₀₁ ⋊ C₄
Hall	C₄₇×C₁₀₁ ⋊ Dic₃
Hall	C₃×C₁₀₁ ⋊ Dic₄₇
Hall	C₃×C₄₇ ⋊ Dic₁₀₁
Sylow	C₁₀₁ ⋊ Dic₁₄₁
Sylow	C₄₇ ⋊ Dic₃₀₃
Sylow	C₃ ⋊ Dic₄₇₄₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₂₄₁
commutator, hypocenter	C₃×C₄₇×C₁₀₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₄₇×C₁₀₁	C₂
LCS	G ▷ C₃×C₄₇×C₁₀₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 155
transitive degree	μ_t(G) = 56964

A minimal faithful permutation representation — 2 generators on 155 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);(1,3,2,4)(6,7)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(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:

2^2,3,47,101|a^2=b,c^a=c^2,d^a=d^46,e^a=e^100

Conjugacy classes & character table

14244 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, 47, 101]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^46);;
gap> SetConjugate(coll, 5, 1, F.5^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14241 #72290770