Dic₆₉₆₆ #72236369

order	27864
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₉₆₆

Splittings

Hall	C₄₃×C₈₁ ⋊ Q₈
Sylow	C₈₁ ⋊ Dic₈₆
Sylow	C₄₃ ⋊ Dic₁₆₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃₄₈₃
commutator, cosocle	C₂×C₄₃×C₈₁	C₂²
Frattini	C₂×C₂₇	C₂×D₁₂₉
Fitting	C₄×C₄₃×C₈₁	C₂
socle, prime radical	C₂×C₃×C₄₃	C₂×D₂₇
hypercenter	C₄	D₃₄₈₃
hypocenter	C₄₃×C₈₁	Q₈
LCS	G ▷ C₂×C₄₃×C₈₁ ▷ C₄₃×C₈₁
UCS	C₁ ◁ C₂ ◁ C₄

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 27864

A minimal faithful permutation representation — 2 generators on 132 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)

Polycyclic presentation:

2^3,3,43,3^3|a^2=c,b^2=c,d^3=f,f^3=g,g^3=h,b^a=bc,d^a=d^2f^2g^2h^2,e^a=e^42,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

6969 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₉₃₂
maximal normal subgroup of	Dic₁₃₉₃₂

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 43, 3, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 5, 1, F.5^42);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6966 #72236369