Dic₃₉₆₀ #72215315

order	15840
automorphism	Aut(G) order 15206400
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₉₆₀

Splittings

Hall	C₅×C₉×C₁₁ ⋊ Q₃₂
Hall	C₉×C₁₁ ⋊ Dic₄₀
Hall	C₅×C₁₁ ⋊ Dic₇₂
Hall	C₅×C₉ ⋊ Dic₈₈
Sylow	C₁₁ ⋊ Dic₃₆₀
Sylow	C₉ ⋊ Dic₄₄₀
Sylow	C₅ ⋊ Dic₇₉₂

Subgroups

	subgroup	G/H
center	C₂	D₃₉₆₀
commutator, cosocle	C₅×C₈×C₉×C₁₁	C₂²
Frattini	C₃×C₈	C₂×D₁₆₅
Fitting	C₅×C₉×C₁₁×C₁₆	C₂
socle, prime radical	C₂×C₃×C₅×C₁₁	D₂₄
hypercenter	C₁₆	D₄₉₅
hypocenter	C₅×C₉×C₁₁	Q₃₂
LCS	G ▷ C₅×C₈×C₉×C₁₁ ▷ C₄×C₅×C₉×C₁₁ ▷ C₂×C₅×C₉×C₁₁ ▷ C₅×C₉×C₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

144 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 15840

A minimal faithful permutation representation — 2 generators on 57 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,41)(35,40)(36,39)(37,38)(43,46)(44,45)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

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

Conjugacy classes & character table

3963 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₉₂₀, Dic₁₁₈₈₀
maximal normal subgroup of	Dic₇₉₂₀
maximal quotient of	Dic₁₁₈₈₀

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 5, 11, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^10);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);