Dic₁₁₈₆₈ #72279133

order	47472
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₁₈₆₈

Splittings

Hall	C₃×C₂₃×C₄₃ ⋊ Q₁₆
Hall	C₂₃×C₄₃ ⋊ Dic₁₂
Hall	C₃×C₄₃ ⋊ Dic₉₂
Hall	C₃×C₂₃ ⋊ Dic₁₇₂
Sylow	C₄₃ ⋊ Dic₂₇₆
Sylow	C₂₃ ⋊ Dic₅₁₆
Sylow	C₃ ⋊ Dic₃₉₅₆

Subgroups

	subgroup	G/H
center	C₂	D₁₁₈₆₈
commutator, cosocle	C₃×C₄×C₂₃×C₄₃	C₂²
Frattini	C₄	C₂×D₂₉₆₇
Fitting	C₃×C₈×C₂₃×C₄₃	C₂
socle, prime radical	C₂×C₃×C₂₃×C₄₃	D₄
hypercenter	C₈	D₂₉₆₇
hypocenter	C₃×C₂₃×C₄₃	Q₁₆
LCS	G ▷ C₃×C₄×C₂₃×C₄₃ ▷ C₂×C₃×C₂₃×C₄₃ ▷ C₃×C₂₃×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 47472

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

Polycyclic presentation:

2^4,3,23,43|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^22,g^a=g^42

Conjugacy classes & character table

11871 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, 23, 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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^22);;
gap> SetConjugate(coll, 7, 1, F.7^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic11868 #72279133