Dic₁₁₆₀ #72211795

order	4640
automorphism	Aut(G) order 2078720
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₁₆₀

Splittings

Hall	C₅×C₂₉ ⋊ Q₃₂
Sylow	C₂₉ ⋊ Dic₄₀
Sylow	C₅ ⋊ Dic₂₃₂

Subgroups

	subgroup	G/H
center	C₂	D₁₁₆₀
commutator, cosocle	C₅×C₈×C₂₉	C₂²
Frattini	C₈	C₂×D₁₄₅
Fitting	C₅×C₁₆×C₂₉	C₂
socle, prime radical	C₂×C₅×C₂₉	D₈
hypercenter	C₁₆	D₁₄₅
hypocenter	C₅×C₂₉	Q₃₂
LCS	G ▷ C₅×C₈×C₂₉ ▷ C₄×C₅×C₂₉ ▷ C₂×C₅×C₂₉ ▷ C₅×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 4640

A minimal faithful permutation representation — 2 generators on 66 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)(38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66);(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)(34,37)(35,36)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

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

Conjugacy classes & character table

1163 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₂₃₂₀, Dic₃₄₈₀, Dic₅₈₀₀, …
maximal normal subgroup of	Dic₂₃₂₀
maximal quotient of	Dic₃₄₈₀, Dic₅₈₀₀, Dic₈₁₂₀, …

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, 2, 5, 29]);;
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> 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.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1160 #72211795