Dic₁₇₁₆ #72212155

order	6864
automorphism	Aut(G) order 3294720
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₇₁₆

Splittings

Hall	C₃×C₁₁×C₁₃ ⋊ Q₁₆
Hall	C₁₁×C₁₃ ⋊ Dic₁₂
Hall	C₃×C₁₃ ⋊ Dic₄₄
Hall	C₃×C₁₁ ⋊ Dic₅₂
Sylow	C₁₃ ⋊ Dic₁₃₂
Sylow	C₁₁ ⋊ Dic₁₅₆
Sylow	C₃ ⋊ Dic₅₇₂

Subgroups

	subgroup	G/H
center	C₂	D₁₇₁₆
commutator, cosocle	C₃×C₄×C₁₁×C₁₃	C₂²
Frattini	C₄	C₂×D₄₂₉
Fitting	C₃×C₈×C₁₁×C₁₃	C₂
socle, prime radical	C₂×C₃×C₁₁×C₁₃	D₄
hypercenter	C₈	D₄₂₉
hypocenter	C₃×C₁₁×C₁₃	Q₁₆
LCS	G ▷ C₃×C₄×C₁₁×C₁₃ ▷ C₂×C₃×C₁₁×C₁₃ ▷ C₃×C₁₁×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 6864

A minimal faithful permutation representation — 2 generators on 43 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,30)(22,29)(23,28)(24,27)(25,26)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)

Polycyclic presentation:

2^4,3,11,13|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^10,g^a=g^12

Conjugacy classes & character table

1719 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₃₄₃₂, Dic₅₁₄₈, Dic₈₅₈₀, …
maximal normal subgroup of	Dic₃₄₃₂
maximal quotient of	Dic₅₁₄₈, Dic₈₅₈₀, Dic₁₂₀₁₂

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, 2, 2, 3, 11, 13]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1716 #72212155