Dic₁₄₆₀ #72211960

order	5840
automorphism	Aut(G) order 3363840
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₆₀

Splittings

Hall	C₅×C₇₃ ⋊ Q₁₆
Sylow	C₇₃ ⋊ Dic₂₀
Sylow	C₅ ⋊ Dic₂₉₂

Subgroups

	subgroup	G/H
center	C₂	D₁₄₆₀
commutator, cosocle	C₄×C₅×C₇₃	C₂²
Frattini	C₄	C₂×D₃₆₅
Fitting	C₅×C₈×C₇₃	C₂
socle, prime radical	C₂×C₅×C₇₃	D₄
hypercenter	C₈	D₃₆₅
hypocenter	C₅×C₇₃	Q₁₆
LCS	G ▷ C₄×C₅×C₇₃ ▷ C₂×C₅×C₇₃ ▷ C₅×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 94
transitive degree	μ_t(G) = 5840

A minimal faithful permutation representation — 2 generators on 94 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,21)(19,20)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Polycyclic presentation:

2^4,5,73|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^72

Conjugacy classes & character table

1463 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₂₉₂₀, Dic₄₃₈₀, Dic₇₃₀₀, …
maximal normal subgroup of	Dic₂₉₂₀
maximal quotient of	Dic₄₃₈₀, Dic₇₃₀₀, Dic₁₀₂₂₀, …

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 5, 73]);;
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.6^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1460 #72211960