D₃₉₈₄ #72211264

order	7968
automorphism	Aut(G) order 5227008
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₉₈₄

Splittings

Fitting	C₃×C₁₆×C₈₃ ⋊ C₂
Hall	C₃×C₈₃ ⋊ D₁₆
Sylow	C₈₃ ⋊ D₄₈
Sylow	C₃ ⋊ D₁₃₂₈

Subgroups

	subgroup	G/H
center	C₂	D₁₉₉₂
commutator, cosocle	C₃×C₈×C₈₃	C₂²
Frattini	C₈	C₂×D₂₄₉
Fitting	C₃×C₁₆×C₈₃	C₂
socle	C₂×C₃×C₈₃	D₈
hypercenter	C₁₆	D₂₄₉
hypocenter	C₃×C₈₃	D₁₆
LCS	G ▷ C₃×C₈×C₈₃ ▷ C₃×C₄×C₈₃ ▷ C₂×C₃×C₈₃ ▷ C₃×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 102
transitive degree	μ_t(G) = 3984

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

Polycyclic presentation:

2^5,3,83|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^82

Conjugacy classes & character table

1995 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇₉₆₈, D₁₁₉₅₂, D₁₉₉₂₀, …
maximal normal subgroup of	D₇₉₆₈
maximal quotient of	D₇₉₆₈, Dic₃₉₈₄, D₁₁₉₅₂, …

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, 3, 83]);;
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.7^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D3984 #72211264