D₁₃₄₀ #72210175

order	2680
automorphism	Aut(G) order 707520
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₄₀

Splittings

Fitting	C₄×C₅×C₆₇ ⋊ C₂
Hall	C₅×C₆₇ ⋊ D₄
Sylow	C₆₇ ⋊ D₂₀
Sylow	C₅ ⋊ D₂₆₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₃₅
commutator, socle, cosocle	C₂×C₅×C₆₇	C₂²
Fitting	C₄×C₅×C₆₇	C₂
hypercenter	C₄	D₃₃₅
hypocenter	C₅×C₆₇	D₄
LCS	G ▷ C₂×C₅×C₆₇ ▷ C₅×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 76
transitive degree	μ_t(G) = 1340

A minimal faithful permutation representation — 2 generators on 76 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);(2,4)(6,9)(7,8)(11,76)(12,75)(13,74)(14,73)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(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:

2^3,5,67|b^2=c,b^a=bc,d^a=d^4,e^a=e^66

Conjugacy classes & character table

673 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₆₈₀, D₄₀₂₀, D₆₇₀₀, …
maximal normal subgroup of	D₂₆₈₀
maximal quotient of	D₂₆₈₀, D₄₀₂₀, Dic₁₃₄₀, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 67]);;
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.5^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1340 #72210175