D₁₉₈₅₆ #72274185

order	39712
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₉₈₅₆

Splittings

Fitting	C₁₆×C₁₇×C₇₃ ⋊ C₂
Hall	C₁₇×C₇₃ ⋊ D₁₆
Sylow	C₇₃ ⋊ D₂₇₂
Sylow	C₁₇ ⋊ D₁₁₆₈

Subgroups

	subgroup	G/H
center	C₂	D₉₉₂₈
commutator, cosocle	C₈×C₁₇×C₇₃	C₂²
Frattini	C₈	C₂×D₁₂₄₁
Fitting	C₁₆×C₁₇×C₇₃	C₂
socle	C₂×C₁₇×C₇₃	D₈
hypercenter	C₁₆	D₁₂₄₁
hypocenter	C₁₇×C₇₃	D₁₆
LCS	G ▷ C₈×C₁₇×C₇₃ ▷ C₄×C₁₇×C₇₃ ▷ C₂×C₁₇×C₇₃ ▷ C₁₇×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 106
transitive degree	μ_t(G) = 19856

A minimal faithful permutation representation — 2 generators on 106 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)(34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)

Polycyclic presentation:

2^5,17,73|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^16,g^a=g^72

Conjugacy classes & character table

9931 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, 2, 17, 73]);;
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.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D19856 #72274185