Dic₉₈₀ #72211540

order	3920
automorphism	Aut(G) order 1317120
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₉₈₀

Splittings

Hall	C₅×C₄₉ ⋊ Q₁₆
Sylow	C₄₉ ⋊ Dic₂₀
Sylow	C₅ ⋊ Dic₁₉₆

Subgroups

	subgroup	G/H
center	C₂	D₉₈₀
commutator, cosocle	C₄×C₅×C₄₉	C₂²
Frattini	C₄×C₇	C₂×D₃₅
Fitting	C₅×C₈×C₄₉	C₂
socle, prime radical	C₂×C₅×C₇	D₂₈
hypercenter	C₈	D₂₄₅
hypocenter	C₅×C₄₉	Q₁₆
LCS	G ▷ C₄×C₅×C₄₉ ▷ C₂×C₅×C₄₉ ▷ C₅×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 3920

A minimal faithful permutation representation — 2 generators on 70 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,21)(19,20)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Polycyclic presentation:

2^4,5,7^2|a^2=d,b^2=cd,c^2=d,f^7=g,b^a=bc,c^a=cd,e^a=e^4,f^a=f^6g^6,g^a=g^6

Conjugacy classes & character table

983 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₁₉₆₀, Dic₂₉₄₀, Dic₄₉₀₀, …
maximal normal subgroup of	Dic₁₉₆₀
maximal quotient of	Dic₂₉₄₀, Dic₄₉₀₀, Dic₆₈₆₀, …

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

Dic980 #72211540