Dic₅₄₂₇ #72217475

order	21708
automorphism	Aut(G) order 38683656
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₄₂₇

Splittings

derived, Hall	C₆₇×C₈₁ ⋊ C₄
Sylow	C₈₁ ⋊ Dic₆₇
Sylow	C₆₇ ⋊ Dic₈₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₄₂₇
commutator, hypocenter	C₆₇×C₈₁	C₄
Frattini	C₂×C₂₇	D₂₀₁
Fitting, cosocle	C₂×C₆₇×C₈₁	C₂
socle, prime radical	C₂×C₃×C₆₇	D₂₇
LCS	G ▷ C₆₇×C₈₁
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 152
transitive degree	μ_t(G) = 21708

A minimal faithful permutation representation — 2 generators on 152 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);(1,3,2,4)(6,85)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(87,152)(88,151)(89,150)(90,149)(91,148)(92,147)(93,146)(94,145)(95,144)(96,143)(97,142)(98,141)(99,140)(100,139)(101,138)(102,137)(103,136)(104,135)(105,134)(106,133)(107,132)(108,131)(109,130)(110,129)(111,128)(112,127)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)

Polycyclic presentation:

2^2,3,67,3^3|a^2=b,c^3=e,e^3=f,f^3=g,c^a=c^2e^2f^2g^2,d^a=d^66,e^a=e^2f^2g^2,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

5430 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₈₅₄, Dic₁₆₂₈₁
maximal normal subgroup of	Dic₁₀₈₅₄
maximal quotient of	Dic₁₆₂₈₁

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 3, 67, 3, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.5^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 4, 1, F.4^66);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5427 #72217475