Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,6,27}

Atlas Canonical Name {2,6,27}*648

Overview

Group
SmallGroup(648,298)
Rank
4
Schläfli Type
{2,6,27}
Vertices, edges, …
2, 6, 81, 27
Order of s0s1s2s3
54
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

Covers minimal covers in bold

2-fold

3-fold

Representations

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