D₁₂₉₆ #72210155

order	2592
automorphism	Aut(G) order 559872
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₂₉₆

Splittings

Fitting	C₁₆×C₈₁ ⋊ C₂
Sylow	C₈₁ ⋊ D₁₆

Subgroups

	subgroup	G/H
center	C₂	D₆₄₈
commutator, cosocle	C₈×C₈₁	C₂²
Frattini	C₈×C₂₇	C₂×D₃
Fitting	C₁₆×C₈₁	C₂
socle	C₂×C₃	D₂₁₆
hypercenter	C₁₆	D₈₁
hypocenter	C₈₁	D₁₆
LCS	G ▷ C₈×C₈₁ ▷ C₄×C₈₁ ▷ C₂×C₈₁ ▷ C₈₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 1296

A minimal faithful permutation representation — 2 generators on 97 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,97)(19,96)(20,95)(21,94)(22,93)(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:

2^5,3^4|b^2=cd,c^2=de,d^2=e,f^3=g,g^3=h,h^3=i,b^a=bc,c^a=cd,d^a=de,f^a=f^2g^2h^2i^2,g^a=g^2h^2i^2,h^a=h^2i^2,i^a=i^2

Conjugacy classes & character table

651 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₅₉₂, D₃₈₈₈, D₆₄₈₀, …
maximal normal subgroup of	D₂₅₉₂
maximal quotient of	D₂₅₉₂, D₃₈₈₈, Dic₁₂₉₆, …

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", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1296 #72210155