Dic₃₁₆₈ #72213487

order	12672
automorphism	Aut(G) order 12165120
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₁₆₈

Splittings

Hall	C₉×C₁₁ ⋊ Q₁₂₈
Sylow	C₁₁ ⋊ Dic₂₈₈
Sylow	C₉ ⋊ Dic₃₅₂

Subgroups

	subgroup	G/H
center	C₂	D₃₁₆₈
commutator, cosocle	C₉×C₁₁×C₃₂	C₂²
Frattini	C₃×C₃₂	C₂×D₃₃
Fitting	C₉×C₁₁×C₆₄	C₂
socle, prime radical	C₂×C₃×C₁₁	D₉₆
hypercenter	C₆₄	D₉₉
hypocenter	C₉×C₁₁	Q₁₂₈
LCS	G ▷ C₉×C₁₁×C₃₂ ▷ C₉×C₁₁×C₁₆ ▷ C₈×C₉×C₁₁ ▷ C₄×C₉×C₁₁ ▷ C₂×C₉×C₁₁ ▷ C₉×C₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 148
transitive degree	μ_t(G) = 12672

A minimal faithful permutation representation — 2 generators on 148 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)(65,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66)(129,130,131,132,133,134,135,136,137)(138,139,140,141,142,143,144,145,146,147,148);(1,65,33,97)(2,66,34,98)(3,67,35,99)(4,68,36,100)(5,69,37,101)(6,70,38,102)(7,71,39,103)(8,72,40,104)(9,73,41,105)(10,74,42,106)(11,75,43,107)(12,76,44,108)(13,77,45,109)(14,78,46,110)(15,79,47,111)(16,80,48,112)(17,81,49,113)(18,82,50,114)(19,83,51,115)(20,84,52,116)(21,85,53,117)(22,86,54,118)(23,87,55,119)(24,88,56,120)(25,89,57,121)(26,90,58,122)(27,91,59,123)(28,92,60,124)(29,93,61,125)(30,94,62,126)(31,95,63,127)(32,96,64,128)(130,137)(131,136)(132,135)(133,134)(139,148)(140,147)(141,146)(142,145)(143,144)

Polycyclic presentation:

2^7,3,11,3|a^2=g,b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=g,h^3=j,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^2j^2,i^a=i^10,j^a=j^2

Conjugacy classes & character table

3171 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₃₃₆, Dic₉₅₀₄, Dic₁₅₈₄₀
maximal normal subgroup of	Dic₆₃₃₆
maximal quotient of	Dic₉₅₀₄, Dic₁₅₈₄₀

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", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 11, 3]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.10^2);;
gap> SetConjugate(coll, 9, 1, F.9^10);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3168 #72213487