Dic₈₉₁ #72211124

order	3564
automorphism	Aut(G) order 962280
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₈₉₁

Splittings

derived, Hall	C₁₁×C₈₁ ⋊ C₄
Sylow	C₈₁ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₈₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₈₉₁
commutator, hypocenter	C₁₁×C₈₁	C₄
Frattini	C₂×C₂₇	D₃₃
Fitting, cosocle	C₂×C₁₁×C₈₁	C₂
socle, prime radical	C₂×C₃×C₁₁	D₂₇
LCS	G ▷ C₁₁×C₈₁
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 3564

A minimal faithful permutation representation — 2 generators on 96 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);(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)(87,96)(88,95)(89,94)(90,93)(91,92)

Polycyclic presentation:

2^2,3,11,3^3|a^2=b,c^3=e,e^3=f,f^3=g,c^a=c^2e^2f^2g^2,d^a=d^10,e^a=e^2f^2g^2,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

894 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₁₇₈₂, Dic₂₆₇₃, Dic₄₄₅₅, …
maximal normal subgroup of	Dic₁₇₈₂
maximal quotient of	Dic₂₆₇₃, Dic₄₄₅₅, Dic₆₂₃₇, …

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, 11, 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.4^10);;
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);

Dic891 #72211124