D₁₄₁₆ #72210195

order	2832
automorphism	Aut(G) order 657024
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₁₆

Splittings

Fitting	C₃×C₈×C₅₉ ⋊ C₂
Hall	C₃×C₅₉ ⋊ D₈
Sylow	C₅₉ ⋊ D₂₄
Sylow	C₃ ⋊ D₄₇₂

Subgroups

	subgroup	G/H
center	C₂	D₇₀₈
commutator, cosocle	C₃×C₄×C₅₉	C₂²
Frattini	C₄	C₂×D₁₇₇
Fitting	C₃×C₈×C₅₉	C₂
socle	C₂×C₃×C₅₉	D₄
hypercenter	C₈	D₁₇₇
hypocenter	C₃×C₅₉	D₈
LCS	G ▷ C₃×C₄×C₅₉ ▷ C₂×C₃×C₅₉ ▷ C₃×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 1416

A minimal faithful permutation representation — 2 generators on 70 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);(2,8)(3,7)(4,6)(10,11)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)

Polycyclic presentation:

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

Conjugacy classes & character table

711 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₈₃₂, D₄₂₄₈, D₇₀₈₀, …
maximal normal subgroup of	D₂₈₃₂
maximal quotient of	D₂₈₃₂, D₄₂₄₈, Dic₁₄₁₆, …

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 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.6^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1416 #72210195