Dic₁₄₂₀ #72211945

order	5680
automorphism	Aut(G) order 3180800
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₂₀

Splittings

Hall	C₅×C₇₁ ⋊ Q₁₆
Sylow	C₇₁ ⋊ Dic₂₀
Sylow	C₅ ⋊ Dic₂₈₄

Subgroups

	subgroup	G/H
center	C₂	D₁₄₂₀
commutator, cosocle	C₄×C₅×C₇₁	C₂²
Frattini	C₄	C₂×D₃₅₅
Fitting	C₅×C₈×C₇₁	C₂
socle, prime radical	C₂×C₅×C₇₁	D₄
hypercenter	C₈	D₃₅₅
hypocenter	C₅×C₇₁	Q₁₆
LCS	G ▷ C₄×C₅×C₇₁ ▷ C₂×C₅×C₇₁ ▷ C₅×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 92
transitive degree	μ_t(G) = 5680

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

Polycyclic presentation:

2^4,5,71|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^70

Conjugacy classes & character table

1423 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₂₈₄₀, Dic₄₂₆₀, Dic₇₁₀₀, …
maximal normal subgroup of	Dic₂₈₄₀
maximal quotient of	Dic₄₂₆₀, Dic₇₁₀₀, Dic₉₉₄₀, …

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, 71]);;
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.6^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1420 #72211945