D₁₈₄₀₈ #72273554

order	36816
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₈₄₀₈

Splittings

Fitting	C₃×C₈×C₁₃×C₅₉ ⋊ C₂
Hall	C₃×C₁₃×C₅₉ ⋊ D₈
Hall	C₁₃×C₅₉ ⋊ D₂₄
Hall	C₃×C₅₉ ⋊ D₁₀₄
Sylow	C₅₉ ⋊ D₃₁₂
Hall	C₃×C₁₃ ⋊ D₄₇₂
Sylow	C₁₃ ⋊ D₁₄₁₆
Sylow	C₃ ⋊ D₆₁₃₆

Subgroups

	subgroup	G/H
center	C₂	D₉₂₀₄
commutator, cosocle	C₃×C₄×C₁₃×C₅₉	C₂²
Frattini	C₄	C₂×D₂₃₀₁
Fitting	C₃×C₈×C₁₃×C₅₉	C₂
socle	C₂×C₃×C₁₃×C₅₉	D₄
hypercenter	C₈	D₂₃₀₁
hypocenter	C₃×C₁₃×C₅₉	D₈
LCS	G ▷ C₃×C₄×C₁₃×C₅₉ ▷ C₂×C₃×C₁₃×C₅₉ ▷ C₃×C₁₃×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 18408

A minimal faithful permutation representation — 2 generators on 83 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);(2,8)(3,7)(4,6)(10,11)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2^4,3,13,59|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^12,g^a=g^58

Conjugacy classes & character table

9207 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 13, 59]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D18408 #72273554