Dic₅₄₀ #72210340

order	2160
automorphism	Aut(G) order 311040
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₄₀

Splittings

Hall	C₅×C₂₇ ⋊ Q₁₆
Sylow	C₂₇ ⋊ Dic₂₀
Sylow	C₅ ⋊ Dic₁₀₈

Subgroups

	subgroup	G/H
center	C₂	D₅₄₀
commutator, cosocle	C₄×C₅×C₂₇	C₂²
Frattini	C₄×C₉	C₂×D₁₅
Fitting	C₅×C₈×C₂₇	C₂
socle, prime radical	C₂×C₃×C₅	D₃₆
hypercenter	C₈	D₁₃₅
hypocenter	C₅×C₂₇	Q₁₆
LCS	G ▷ C₄×C₅×C₂₇ ▷ C₂×C₅×C₂₇ ▷ C₅×C₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 2160

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

Polycyclic presentation:

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

Conjugacy classes & character table

543 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₁₀₈₀, Dic₁₆₂₀, Dic₂₇₀₀, …
maximal normal subgroup of	Dic₁₀₈₀
maximal quotient of	Dic₁₆₂₀, Dic₂₇₀₀, Dic₃₇₈₀, …

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

Dic540 #72210340