Dic₁₆₃₄₀ #72301591

order	65360
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₆₃₄₀

Splittings

Hall	C₅×C₁₉×C₄₃ ⋊ Q₁₆
Hall	C₁₉×C₄₃ ⋊ Dic₂₀
Hall	C₅×C₄₃ ⋊ Dic₇₆
Hall	C₅×C₁₉ ⋊ Dic₁₇₂
Sylow	C₄₃ ⋊ Dic₃₈₀
Sylow	C₁₉ ⋊ Dic₈₆₀
Sylow	C₅ ⋊ Dic₃₂₆₈

Subgroups

	subgroup	G/H
center	C₂	D₁₆₃₄₀
commutator, cosocle	C₄×C₅×C₁₉×C₄₃	C₂²
Frattini	C₄	C₂×D₄₀₈₅
Fitting	C₅×C₈×C₁₉×C₄₃	C₂
socle, prime radical	C₂×C₅×C₁₉×C₄₃	D₄
hypercenter	C₈	D₄₀₈₅
hypocenter	C₅×C₁₉×C₄₃	Q₁₆
LCS	G ▷ C₄×C₅×C₁₉×C₄₃ ▷ C₂×C₅×C₁₉×C₄₃ ▷ C₅×C₁₉×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 65360

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

Polycyclic presentation:

2^4,5,19,43|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^18,g^a=g^42

Conjugacy classes & character table

16343 conjugacy classes. The character table is too large to display.

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, 19, 43]);;
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.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic16340 #72301591