D₁₀₅₂₈ #72215430

order	21056
automorphism	Aut(G) order 46491648
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₅₂₈

Splittings

Fitting	C₇×C₃₂×C₄₇ ⋊ C₂
Hall	C₇×C₄₇ ⋊ D₃₂
Sylow	C₄₇ ⋊ D₂₂₄
Sylow	C₇ ⋊ D₁₅₀₄

Subgroups

	subgroup	G/H
center	C₂	D₅₂₆₄
commutator, cosocle	C₇×C₁₆×C₄₇	C₂²
Frattini	C₁₆	C₂×D₃₂₉
Fitting	C₇×C₃₂×C₄₇	C₂
socle	C₂×C₇×C₄₇	D₁₆
hypercenter	C₃₂	D₃₂₉
hypocenter	C₇×C₄₇	D₃₂
LCS	G ▷ C₇×C₁₆×C₄₇ ▷ C₇×C₈×C₄₇ ▷ C₄×C₇×C₄₇ ▷ C₂×C₇×C₄₇ ▷ C₇×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 86
transitive degree	μ_t(G) = 10528

A minimal faithful permutation representation — 2 generators on 86 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);(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)(34,39)(35,38)(36,37)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2^6,7,47|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^6,h^a=h^46

Conjugacy classes & character table

5267 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₁₀₅₆, D₃₁₅₈₄
maximal normal subgroup of	D₂₁₀₅₆
maximal quotient of	Dic₁₀₅₂₈, D₂₁₀₅₆, D₃₁₅₈₄

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, 47]);;
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> 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);;
gap> SetConjugate(coll, 8, 1, F.8^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D10528 #72215430