D₁₉₈₂₄ #72274169

order	39648
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₉₈₂₄

Splittings

Fitting	C₃×C₇×C₁₆×C₅₉ ⋊ C₂
Hall	C₃×C₇×C₅₉ ⋊ D₁₆
Hall	C₇×C₅₉ ⋊ D₄₈
Hall	C₃×C₅₉ ⋊ D₁₁₂
Sylow	C₅₉ ⋊ D₃₃₆
Hall	C₃×C₇ ⋊ D₉₄₄
Sylow	C₇ ⋊ D₂₈₃₂
Sylow	C₃ ⋊ D₆₆₀₈

Subgroups

	subgroup	G/H
center	C₂	D₉₉₁₂
commutator, cosocle	C₃×C₇×C₈×C₅₉	C₂²
Frattini	C₈	C₂×D₁₂₃₉
Fitting	C₃×C₇×C₁₆×C₅₉	C₂
socle	C₂×C₃×C₇×C₅₉	D₈
hypercenter	C₁₆	D₁₂₃₉
hypocenter	C₃×C₇×C₅₉	D₁₆
LCS	G ▷ C₃×C₇×C₈×C₅₉ ▷ C₃×C₄×C₇×C₅₉ ▷ C₂×C₃×C₇×C₅₉ ▷ C₃×C₇×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 19824

A minimal faithful permutation representation — 2 generators on 85 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,26)(22,25)(23,24)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2^5,3,7,59|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^6,h^a=h^58

Conjugacy classes & character table

9915 conjugacy classes. The character table is too large to display.

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, 3, 7, 59]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
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);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D19824 #72274169