D₁₀₆₅₆ #72215560

order	21312
automorphism	Aut(G) order 36827136
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₆₅₆

Splittings

Fitting	C₉×C₃₂×C₃₇ ⋊ C₂
Hall	C₉×C₃₇ ⋊ D₃₂
Sylow	C₃₇ ⋊ D₂₈₈
Sylow	C₉ ⋊ D₁₁₈₄

Subgroups

	subgroup	G/H
center	C₂	D₅₃₂₈
commutator, cosocle	C₉×C₁₆×C₃₇	C₂²
Frattini	C₃×C₁₆	C₂×D₁₁₁
Fitting	C₉×C₃₂×C₃₇	C₂
socle	C₂×C₃×C₃₇	D₄₈
hypercenter	C₃₂	D₃₃₃
hypocenter	C₉×C₃₇	D₃₂
LCS	G ▷ C₉×C₁₆×C₃₇ ▷ C₈×C₉×C₃₇ ▷ C₄×C₉×C₃₇ ▷ C₂×C₉×C₃₇ ▷ C₉×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 78
transitive degree	μ_t(G) = 10656

A minimal faithful permutation representation — 2 generators on 78 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);(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)(34,41)(35,40)(36,39)(37,38)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)

Polycyclic presentation:

2^6,3,37,3|b^2=cd,c^2=de,d^2=ef,e^2=f,g^3=i,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2i^2,h^a=h^36,i^a=i^2

Conjugacy classes & character table

5331 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₁₃₁₂, D₃₁₉₆₈
maximal normal subgroup of	D₂₁₃₁₂
maximal quotient of	Dic₁₀₆₅₆, D₂₁₃₁₂, D₃₁₉₆₈

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, 2, 3, 37, 3]);;
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.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, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^36);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D10656 #72215560