Dic₁₄₂₇₁ #72290816

order	57084
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₂₇₁

Splittings

derived, Hall	C₃×C₆₇×C₇₁ ⋊ C₄
Hall	C₆₇×C₇₁ ⋊ Dic₃
Hall	C₃×C₇₁ ⋊ Dic₆₇
Hall	C₃×C₆₇ ⋊ Dic₇₁
Sylow	C₇₁ ⋊ Dic₂₀₁
Sylow	C₆₇ ⋊ Dic₂₁₃
Sylow	C₃ ⋊ Dic₄₇₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₂₇₁
commutator, hypocenter	C₃×C₆₇×C₇₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₆₇×C₇₁	C₂
LCS	G ▷ C₃×C₆₇×C₇₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 145
transitive degree	μ_t(G) = 57084

A minimal faithful permutation representation — 2 generators on 145 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);(1,3,2,4)(6,7)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(76,145)(77,144)(78,143)(79,142)(80,141)(81,140)(82,139)(83,138)(84,137)(85,136)(86,135)(87,134)(88,133)(89,132)(90,131)(91,130)(92,129)(93,128)(94,127)(95,126)(96,125)(97,124)(98,123)(99,122)(100,121)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)

Polycyclic presentation:

2^2,3,67,71|a^2=b,c^a=c^2,d^a=d^66,e^a=e^70

Conjugacy classes & character table

14274 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, 67, 71]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^66);;
gap> SetConjugate(coll, 5, 1, F.5^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14271 #72290816