D₂₆₇₈₄ #72290127

order	53568
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₇₈₄

Splittings

Fitting	C₂₇×C₃₁×C₃₂ ⋊ C₂
Hall	C₂₇×C₃₁ ⋊ D₃₂
Sylow	C₃₁ ⋊ D₈₆₄
Sylow	C₂₇ ⋊ D₉₉₂

Subgroups

	subgroup	G/H
center	C₂	D₁₃₃₉₂
commutator, cosocle	C₁₆×C₂₇×C₃₁	C₂²
Frattini	C₉×C₁₆	C₂×D₉₃
Fitting	C₂₇×C₃₁×C₃₂	C₂
socle	C₂×C₃×C₃₁	D₁₄₄
hypercenter	C₃₂	D₈₃₇
hypocenter	C₂₇×C₃₁	D₃₂
LCS	G ▷ C₁₆×C₂₇×C₃₁ ▷ C₈×C₂₇×C₃₁ ▷ C₄×C₂₇×C₃₁ ▷ C₂×C₂₇×C₃₁ ▷ C₂₇×C₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 90
transitive degree	μ_t(G) = 26784

A minimal faithful permutation representation — 2 generators on 90 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)(60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2^6,3,31,3^2|b^2=cd,c^2=de,d^2=ef,e^2=f,g^3=i,i^3=j,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2i^2j^2,h^a=h^30,i^a=i^2j^2,j^a=j^2

Conjugacy classes & character table

13395 conjugacy classes. The character table is too large to display.

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", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 31, 3, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.9^2*F.10^2);;
gap> SetConjugate(coll, 8, 1, F.8^30);;
gap> SetConjugate(coll, 9, 1, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26784 #72290127