D₁₅₆₈ #72210243

order	3136
automorphism	Aut(G) order 1053696
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₅₆₈

Splittings

Fitting	C₃₂×C₄₉ ⋊ C₂
Sylow	C₄₉ ⋊ D₃₂

Subgroups

	subgroup	G/H
center	C₂	D₇₈₄
commutator, cosocle	C₁₆×C₄₉	C₂²
Frattini	C₇×C₁₆	C₂×D₇
Fitting	C₃₂×C₄₉	C₂
socle	C₂×C₇	D₁₁₂
hypercenter	C₃₂	D₄₉
hypocenter	C₄₉	D₃₂
LCS	G ▷ C₁₆×C₄₉ ▷ C₈×C₄₉ ▷ C₄×C₄₉ ▷ C₂×C₄₉ ▷ C₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

51 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 1568

A minimal faithful permutation representation — 2 generators on 81 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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(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:

2^6,7^2|b^2=cd,c^2=de,d^2=ef,e^2=f,g^7=h,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^6h^6,h^a=h^6

Conjugacy classes & character table

787 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃₁₃₆, D₄₇₀₄, D₇₈₄₀, …
maximal normal subgroup of	D₃₁₃₆
maximal quotient of	D₃₁₃₆, D₄₇₀₄, Dic₁₅₆₈, …

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

D1568 #72210243