Dic₁₆₃₅₆ #72301623

order	65424
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₆₃₅₆

Splittings

Hall	C₃×C₂₉×C₄₇ ⋊ Q₁₆
Hall	C₂₉×C₄₇ ⋊ Dic₁₂
Hall	C₃×C₄₇ ⋊ Dic₁₁₆
Hall	C₃×C₂₉ ⋊ Dic₁₈₈
Sylow	C₄₇ ⋊ Dic₃₄₈
Sylow	C₂₉ ⋊ Dic₅₆₄
Sylow	C₃ ⋊ Dic₅₄₅₂

Subgroups

	subgroup	G/H
center	C₂	D₁₆₃₅₆
commutator, cosocle	C₃×C₄×C₂₉×C₄₇	C₂²
Frattini	C₄	C₂×D₄₀₈₉
Fitting	C₃×C₈×C₂₉×C₄₇	C₂
socle, prime radical	C₂×C₃×C₂₉×C₄₇	D₄
hypercenter	C₈	D₄₀₈₉
hypocenter	C₃×C₂₉×C₄₇	Q₁₆
LCS	G ▷ C₃×C₄×C₂₉×C₄₇ ▷ C₂×C₃×C₂₉×C₄₇ ▷ C₃×C₂₉×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 65424

A minimal faithful permutation representation — 2 generators on 95 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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)

Polycyclic presentation:

2^4,3,29,47|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^28,g^a=g^46

Conjugacy classes & character table

16359 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, 3, 29, 47]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> SetConjugate(coll, 7, 1, F.7^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic16356 #72301623