Dic₅₀₄₀ #72216626

order	20160
automorphism	Aut(G) order 23224320
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₀₄₀

Splittings

Hall	C₅×C₇×C₉ ⋊ Q₆₄
Hall	C₇×C₉ ⋊ Dic₈₀
Hall	C₅×C₉ ⋊ Dic₁₁₂
Hall	C₅×C₇ ⋊ Dic₁₄₄
Sylow	C₉ ⋊ Dic₅₆₀
Sylow	C₇ ⋊ Dic₇₂₀
Sylow	C₅ ⋊ Dic₁₀₀₈

Subgroups

	subgroup	G/H
center	C₂	D₅₀₄₀
commutator, cosocle	C₅×C₇×C₉×C₁₆	C₂²
Frattini	C₃×C₁₆	C₂×D₁₀₅
Fitting	C₅×C₇×C₉×C₃₂	C₂
socle, prime radical	C₂×C₃×C₅×C₇	D₄₈
hypercenter	C₃₂	D₃₁₅
hypocenter	C₅×C₇×C₉	Q₆₄
LCS	G ▷ C₅×C₇×C₉×C₁₆ ▷ C₅×C₇×C₈×C₉ ▷ C₄×C₅×C₇×C₉ ▷ C₂×C₅×C₇×C₉ ▷ C₅×C₇×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

180 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 85
transitive degree	μ_t(G) = 20160

A minimal faithful permutation representation — 2 generators on 85 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34)(65,66,67,68,69,70,71,72,73)(74,75,76,77,78)(79,80,81,82,83,84,85);(1,33,17,49)(2,34,18,50)(3,35,19,51)(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(66,73)(67,72)(68,71)(69,70)(75,78)(76,77)(80,85)(81,84)(82,83)

Polycyclic presentation:

2^6,3,5,7,3|a^2=f,b^2=cd,c^2=de,d^2=ef,e^2=f,g^3=j,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2j^2,h^a=h^4,i^a=i^6,j^a=j^2

Conjugacy classes & character table

5043 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₀₈₀, Dic₁₅₁₂₀
maximal normal subgroup of	Dic₁₀₀₈₀
maximal quotient of	Dic₁₅₁₂₀

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