D₁₃₄₄ #72210166

order	2688
automorphism	Aut(G) order 516096
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₄₄

Splittings

Fitting	C₃×C₇×C₆₄ ⋊ C₂
Hall	C₃×C₇ ⋊ D₆₄
Sylow	C₇ ⋊ D₁₉₂
Sylow	C₃ ⋊ D₄₄₈

Subgroups

	subgroup	G/H
center	C₂	D₆₇₂
commutator, cosocle	C₃×C₇×C₃₂	C₂²
Frattini	C₃₂	C₂×D₂₁
Fitting	C₃×C₇×C₆₄	C₂
socle	C₂×C₃×C₇	D₃₂
hypercenter	C₆₄	D₂₁
hypocenter	C₃×C₇	D₆₄
LCS	G ▷ C₃×C₇×C₃₂ ▷ C₃×C₇×C₁₆ ▷ C₃×C₇×C₈ ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 74
transitive degree	μ_t(G) = 1344

A minimal faithful permutation representation — 2 generators on 74 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);(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(66,67)(69,74)(70,73)(71,72)

Polycyclic presentation:

2^7,3,7|b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^2,i^a=i^6

Conjugacy classes & character table

675 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₆₈₈, D₄₀₃₂, D₆₇₂₀, …
maximal normal subgroup of	D₂₆₈₈
maximal quotient of	D₂₆₈₈, D₄₀₃₂, Dic₁₃₄₄, …

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, 2, 3, 7]);;
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*F.7);;
gap> SetPower(coll, 6, F.7);;
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, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1344 #72210166