Dic₅₈₀ #72210407

order	2320
automorphism	Aut(G) order 519680
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₈₀

Splittings

Hall	C₅×C₂₉ ⋊ Q₁₆
Sylow	C₂₉ ⋊ Dic₂₀
Sylow	C₅ ⋊ Dic₁₁₆

Subgroups

	subgroup	G/H
center	C₂	D₅₈₀
commutator, cosocle	C₄×C₅×C₂₉	C₂²
Frattini	C₄	C₂×D₁₄₅
Fitting	C₅×C₈×C₂₉	C₂
socle, prime radical	C₂×C₅×C₂₉	D₄
hypercenter	C₈	D₁₄₅
hypocenter	C₅×C₂₉	Q₁₆
LCS	G ▷ C₄×C₅×C₂₉ ▷ C₂×C₅×C₂₉ ▷ C₅×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 2320

A minimal faithful permutation representation — 2 generators on 50 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,21)(19,20)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)

Polycyclic presentation:

2^4,5,29|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^28

Conjugacy classes & character table

583 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₁₁₆₀, Dic₁₇₄₀, Dic₂₉₀₀, …
maximal normal subgroup of	Dic₁₁₆₀
maximal quotient of	Dic₁₇₄₀, Dic₂₉₀₀, Dic₄₀₆₀, …

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 5, 29]);;
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.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic580 #72210407