Dic₁₉₈₀ #72212357

order	7920
automorphism	Aut(G) order 3801600
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₉₈₀

Splittings

Hall	C₅×C₉×C₁₁ ⋊ Q₁₆
Hall	C₉×C₁₁ ⋊ Dic₂₀
Hall	C₅×C₁₁ ⋊ Dic₃₆
Hall	C₅×C₉ ⋊ Dic₄₄
Sylow	C₁₁ ⋊ Dic₁₈₀
Sylow	C₉ ⋊ Dic₂₂₀
Sylow	C₅ ⋊ Dic₃₉₆

Subgroups

	subgroup	G/H
center	C₂	D₁₉₈₀
commutator, cosocle	C₄×C₅×C₉×C₁₁	C₂²
Frattini	C₃×C₄	C₂×D₁₆₅
Fitting	C₅×C₈×C₉×C₁₁	C₂
socle, prime radical	C₂×C₃×C₅×C₁₁	D₁₂
hypercenter	C₈	D₄₉₅
hypocenter	C₅×C₉×C₁₁	Q₁₆
LCS	G ▷ C₄×C₅×C₉×C₁₁ ▷ C₂×C₅×C₉×C₁₁ ▷ C₅×C₉×C₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 7920

A minimal faithful permutation representation — 2 generators on 41 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,25)(19,24)(20,23)(21,22)(27,30)(28,29)(32,41)(33,40)(34,39)(35,38)(36,37)

Polycyclic presentation:

2^4,3,5,11,3|a^2=d,b^2=cd,c^2=d,e^3=h,b^a=bc,c^a=cd,e^a=e^2h^2,f^a=f^4,g^a=g^10,h^a=h^2

Conjugacy classes & character table

1983 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₃₉₆₀, Dic₅₉₄₀, Dic₉₉₀₀, …
maximal normal subgroup of	Dic₃₉₆₀
maximal quotient of	Dic₅₉₄₀, Dic₉₉₀₀, Dic₁₃₈₆₀

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

Dic1980 #72212357