Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,6,9}

Atlas Canonical Name {2,6,9}*1944c

Overview

Group
SmallGroup(1944,944)
Rank
4
Schläfli Type
{2,6,9}
Vertices, edges, …
2, 54, 243, 81
Order of s0s1s2s3
18
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

27-fold

81-fold

Covers minimal covers in bold

None in this atlas.

Representations

Permutation Representation (GAP) 
s0 := (1,2);;
s1 := ( 6, 9)( 7,10)( 8,11)(12,22)(13,23)(14,21)(15,28)(16,29)(17,27)(18,25)(19,26)(20,24)(33,36)(34,37)(35,38)(39,49)(40,50)(41,48)(42,55)(43,56)(44,54)(45,52)(46,53)(47,51)(60,63)(61,64)(62,65)(66,76)(67,77)(68,75)(69,82)(70,83)(71,81)(72,79)(73,80)(74,78);;
s2 := ( 4, 5)( 6, 9)( 7,11)( 8,10)(12,18)(13,20)(14,19)(16,17)(21,24)(22,26)(23,25)(28,29)(30,79)(31,78)(32,80)(33,76)(34,75)(35,77)(36,82)(37,81)(38,83)(39,58)(40,57)(41,59)(42,64)(43,63)(44,65)(45,61)(46,60)(47,62)(48,73)(49,72)(50,74)(51,70)(52,69)(53,71)(54,67)(55,66)(56,68);;
s3 := ( 3,30)( 4,32)( 5,31)( 6,36)( 7,38)( 8,37)( 9,33)(10,35)(11,34)(12,41)(13,40)(14,39)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,49)(22,48)(23,50)(24,55)(25,54)(26,56)(27,52)(28,51)(29,53)(57,58)(60,64)(61,63)(62,65)(67,68)(69,72)(70,74)(71,73)(75,77)(78,83)(79,82)(80,81);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(83)!(1,2);
s1 := Sym(83)!( 6, 9)( 7,10)( 8,11)(12,22)(13,23)(14,21)(15,28)(16,29)(17,27)(18,25)(19,26)(20,24)(33,36)(34,37)(35,38)(39,49)(40,50)(41,48)(42,55)(43,56)(44,54)(45,52)(46,53)(47,51)(60,63)(61,64)(62,65)(66,76)(67,77)(68,75)(69,82)(70,83)(71,81)(72,79)(73,80)(74,78);
s2 := Sym(83)!( 4, 5)( 6, 9)( 7,11)( 8,10)(12,18)(13,20)(14,19)(16,17)(21,24)(22,26)(23,25)(28,29)(30,79)(31,78)(32,80)(33,76)(34,75)(35,77)(36,82)(37,81)(38,83)(39,58)(40,57)(41,59)(42,64)(43,63)(44,65)(45,61)(46,60)(47,62)(48,73)(49,72)(50,74)(51,70)(52,69)(53,71)(54,67)(55,66)(56,68);
s3 := Sym(83)!( 3,30)( 4,32)( 5,31)( 6,36)( 7,38)( 8,37)( 9,33)(10,35)(11,34)(12,41)(13,40)(14,39)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,49)(22,48)(23,50)(24,55)(25,54)(26,56)(27,52)(28,51)(29,53)(57,58)(60,64)(61,63)(62,65)(67,68)(69,72)(70,74)(71,73)(75,77)(78,83)(79,82)(80,81);
poly := sub<Sym(83)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2 >;