D₁₄₃₁ #72210206

order	2862
automorphism	Aut(G) order 1339416
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₁₄₃₁

Splittings

Fitting, derived, Hall	C₂₇×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ D₂₇
Sylow	C₂₇ ⋊ D₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₇×C₅₃	C₂
Frattini	C₉	D₁₅₉
socle	C₃×C₅₃	D₉
LCS	G ▷ C₂₇×C₅₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 1431

A minimal faithful permutation representation — 2 generators on 80 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);(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2,3,53,3^2|b^3=d,d^3=e,b^a=b^2d^2e^2,c^a=c^52,d^a=d^2e^2,e^a=e^2

Conjugacy classes & character table

717 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₁₄₃₁, D₄₂₉₃, D₇₁₅₅, …
maximal normal subgroup of	C₂×D₁₄₃₁
maximal quotient of	C₂×D₁₄₃₁, D₄₂₉₃, Dic₁₄₃₁, …

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, 3, 53, 3, 3]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.4^2*F.5^2);;
gap> SetConjugate(coll, 3, 1, F.3^52);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.5^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1431 #72210206