D₃₁₀₀₈ #72298769

order	62016
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₁₀₀₈

Splittings

Fitting	C₃×C₁₇×C₁₉×C₃₂ ⋊ C₂
Hall	C₃×C₁₇×C₁₉ ⋊ D₃₂
Hall	C₁₇×C₁₉ ⋊ D₉₆
Hall	C₃×C₁₉ ⋊ D₅₄₄
Hall	C₃×C₁₇ ⋊ D₆₀₈
Sylow	C₁₉ ⋊ D₁₆₃₂
Sylow	C₁₇ ⋊ D₁₈₂₄
Sylow	C₃ ⋊ D₁₀₃₃₆

Subgroups

	subgroup	G/H
center	C₂	D₁₅₅₀₄
commutator, cosocle	C₃×C₁₆×C₁₇×C₁₉	C₂²
Frattini	C₁₆	C₂×D₉₆₉
Fitting	C₃×C₁₇×C₁₉×C₃₂	C₂
socle	C₂×C₃×C₁₇×C₁₉	D₁₆
hypercenter	C₃₂	D₉₆₉
hypocenter	C₃×C₁₇×C₁₉	D₃₂
LCS	G ▷ C₃×C₁₆×C₁₇×C₁₉ ▷ C₃×C₈×C₁₇×C₁₉ ▷ C₃×C₄×C₁₇×C₁₉ ▷ C₂×C₃×C₁₇×C₁₉ ▷ C₃×C₁₇×C₁₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 31008

A minimal faithful permutation representation — 2 generators on 71 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);(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)(34,35)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)

Polycyclic presentation:

2^6,3,17,19|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^2,h^a=h^16,i^a=i^18

Conjugacy classes & character table

15507 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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 17, 19]);;
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.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^16);;
gap> SetConjugate(coll, 9, 1, F.9^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31008 #72298769