Dic₄₀₆₀ #72215433

order	16240
automorphism	Aut(G) order 21826560
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₀₆₀

Splittings

Hall	C₅×C₇×C₂₉ ⋊ Q₁₆
Hall	C₇×C₂₉ ⋊ Dic₂₀
Hall	C₅×C₂₉ ⋊ Dic₂₈
Hall	C₅×C₇ ⋊ Dic₁₁₆
Sylow	C₂₉ ⋊ Dic₁₄₀
Sylow	C₇ ⋊ Dic₅₈₀
Sylow	C₅ ⋊ Dic₈₁₂

Subgroups

	subgroup	G/H
center	C₂	D₄₀₆₀
commutator, cosocle	C₄×C₅×C₇×C₂₉	C₂²
Frattini	C₄	C₂×D₁₀₁₅
Fitting	C₅×C₇×C₈×C₂₉	C₂
socle, prime radical	C₂×C₅×C₇×C₂₉	D₄
hypercenter	C₈	D₁₀₁₅
hypocenter	C₅×C₇×C₂₉	Q₁₆
LCS	G ▷ C₄×C₅×C₇×C₂₉ ▷ C₂×C₅×C₇×C₂₉ ▷ C₅×C₇×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 16240

A minimal faithful permutation representation — 2 generators on 57 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,21)(19,20)(23,28)(24,27)(25,26)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)

Polycyclic presentation:

2^4,5,7,29|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^6,g^a=g^28

Conjugacy classes & character table

4063 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₁₂₀, Dic₁₂₁₈₀
maximal normal subgroup of	Dic₈₁₂₀
maximal quotient of	Dic₁₂₁₈₀

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, 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.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4060 #72215433