D₁₀₆₆₀ #72215546

order	21320
automorphism	Aut(G) order 40934400
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₆₆₀

Splittings

Fitting	C₄×C₅×C₁₃×C₄₁ ⋊ C₂
Hall	C₅×C₁₃×C₄₁ ⋊ D₄
Hall	C₁₃×C₄₁ ⋊ D₂₀
Hall	C₅×C₄₁ ⋊ D₅₂
Hall	C₅×C₁₃ ⋊ D₁₆₄
Sylow	C₄₁ ⋊ D₂₆₀
Sylow	C₁₃ ⋊ D₈₂₀
Sylow	C₅ ⋊ D₂₁₃₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₆₆₅
commutator, socle, cosocle	C₂×C₅×C₁₃×C₄₁	C₂²
Fitting	C₄×C₅×C₁₃×C₄₁	C₂
hypercenter	C₄	D₂₆₆₅
hypocenter	C₅×C₁₃×C₄₁	D₄
LCS	G ▷ C₂×C₅×C₁₃×C₄₁ ▷ C₅×C₁₃×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 10660

A minimal faithful permutation representation — 2 generators on 63 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);(2,4)(6,9)(7,8)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)

Polycyclic presentation:

2^3,5,13,41|b^2=c,b^a=bc,d^a=d^4,e^a=e^12,f^a=f^40

Conjugacy classes & character table

5333 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₁₃₂₀, D₃₁₉₈₀
maximal normal subgroup of	D₂₁₃₂₀
maximal quotient of	Dic₁₀₆₆₀, D₂₁₃₂₀, D₃₁₉₈₀

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, 5, 13, 41]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D10660 #72215546