D₁₃₂₁₆ #72235441

order	26432
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₂₁₆

Splittings

Fitting	C₇×C₃₂×C₅₉ ⋊ C₂
Hall	C₇×C₅₉ ⋊ D₃₂
Sylow	C₅₉ ⋊ D₂₂₄
Sylow	C₇ ⋊ D₁₈₈₈

Subgroups

	subgroup	G/H
center	C₂	D₆₆₀₈
commutator, cosocle	C₇×C₁₆×C₅₉	C₂²
Frattini	C₁₆	C₂×D₄₁₃
Fitting	C₇×C₃₂×C₅₉	C₂
socle	C₂×C₇×C₅₉	D₁₆
hypercenter	C₃₂	D₄₁₃
hypocenter	C₇×C₅₉	D₃₂
LCS	G ▷ C₇×C₁₆×C₅₉ ▷ C₇×C₈×C₅₉ ▷ C₄×C₇×C₅₉ ▷ C₂×C₇×C₅₉ ▷ C₇×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 98
transitive degree	μ_t(G) = 13216

A minimal faithful permutation representation — 2 generators on 98 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)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98);(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)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Polycyclic presentation:

2^6,7,59|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^58

Conjugacy classes & character table

6611 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₆₄₃₂
maximal normal subgroup of	D₂₆₄₃₂
maximal quotient of	Dic₁₃₂₁₆, D₂₆₄₃₂

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, 59]);;
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.8^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D13216 #72235441