Dic₁₃₃₄₀ #72289074

order	53360
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₃₄₀

Splittings

Hall	C₅×C₂₃×C₂₉ ⋊ Q₁₆
Hall	C₂₃×C₂₉ ⋊ Dic₂₀
Hall	C₅×C₂₉ ⋊ Dic₉₂
Hall	C₅×C₂₃ ⋊ Dic₁₁₆
Sylow	C₂₉ ⋊ Dic₄₆₀
Sylow	C₂₃ ⋊ Dic₅₈₀
Sylow	C₅ ⋊ Dic₂₆₆₈

Subgroups

	subgroup	G/H
center	C₂	D₁₃₃₄₀
commutator, cosocle	C₄×C₅×C₂₃×C₂₉	C₂²
Frattini	C₄	C₂×D₃₃₃₅
Fitting	C₅×C₈×C₂₃×C₂₉	C₂
socle, prime radical	C₂×C₅×C₂₃×C₂₉	D₄
hypercenter	C₈	D₃₃₃₅
hypocenter	C₅×C₂₃×C₂₉	Q₁₆
LCS	G ▷ C₄×C₅×C₂₃×C₂₉ ▷ C₂×C₅×C₂₃×C₂₉ ▷ C₅×C₂₃×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 53360

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

Polycyclic presentation:

2^4,5,23,29|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^22,g^a=g^28

Conjugacy classes & character table

13343 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, 23, 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.6^22);;
gap> SetConjugate(coll, 7, 1, F.7^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13340 #72289074