D₄₂₆₀ #72211348

order	8520
automorphism	Aut(G) order 4771200
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₄₂₆₀

Splittings

Fitting	C₃×C₄×C₅×C₇₁ ⋊ C₂
Hall	C₃×C₅×C₇₁ ⋊ D₄
Hall	C₅×C₇₁ ⋊ D₁₂
Hall	C₃×C₇₁ ⋊ D₂₀
Sylow	C₇₁ ⋊ D₆₀
Hall	C₃×C₅ ⋊ D₂₈₄
Sylow	C₅ ⋊ D₈₅₂
Sylow	C₃ ⋊ D₁₄₂₀

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₀₆₅
commutator, socle, cosocle	C₂×C₃×C₅×C₇₁	C₂²
Fitting	C₃×C₄×C₅×C₇₁	C₂
hypercenter	C₄	D₁₀₆₅
hypocenter	C₃×C₅×C₇₁	D₄
LCS	G ▷ C₂×C₃×C₅×C₇₁ ▷ C₃×C₅×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 4260

A minimal faithful permutation representation — 2 generators on 83 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);(2,4)(6,7)(9,12)(10,11)(14,83)(15,82)(16,81)(17,80)(18,79)(19,78)(20,77)(21,76)(22,75)(23,74)(24,73)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2^3,3,5,71|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^70

Conjugacy classes & character table

2133 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₈₅₂₀, D₁₂₇₈₀, D₂₁₃₀₀, …
maximal normal subgroup of	D₈₅₂₀
maximal quotient of	D₈₅₂₀, Dic₄₂₆₀, D₁₂₇₈₀, …

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, 3, 5, 71]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D4260 #72211348