Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,27,2}

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

Overview

Group
SmallGroup(648,298)
Rank
4
Schläfli Type
{6,27,2}
Vertices, edges, …
6, 81, 27, 2
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 := (10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,17)( 5,16)( 6,18)( 7,14)( 8,13)( 9,15)(20,21)(22,26)(23,25)(24,27)(28,67)(29,69)(30,68)(31,64)(32,66)(33,65)(34,71)(35,70)(36,72)(37,58)(38,60)(39,59)(40,55)(41,57)(42,56)(43,62)(44,61)(45,63)(46,76)(47,78)(48,77)(49,73)(50,75)(51,74)(52,80)(53,79)(54,81);;
s2 := ( 1,28)( 2,30)( 3,29)( 4,35)( 5,34)( 6,36)( 7,32)( 8,31)( 9,33)(10,46)(11,48)(12,47)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,37)(20,39)(21,38)(22,44)(23,43)(24,45)(25,41)(26,40)(27,42)(55,58)(56,60)(57,59)(61,62)(64,76)(65,78)(66,77)(67,73)(68,75)(69,74)(70,80)(71,79)(72,81);;
s3 := (82,83);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(83)!(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81);
s1 := Sym(83)!( 1,10)( 2,12)( 3,11)( 4,17)( 5,16)( 6,18)( 7,14)( 8,13)( 9,15)(20,21)(22,26)(23,25)(24,27)(28,67)(29,69)(30,68)(31,64)(32,66)(33,65)(34,71)(35,70)(36,72)(37,58)(38,60)(39,59)(40,55)(41,57)(42,56)(43,62)(44,61)(45,63)(46,76)(47,78)(48,77)(49,73)(50,75)(51,74)(52,80)(53,79)(54,81);
s2 := Sym(83)!( 1,28)( 2,30)( 3,29)( 4,35)( 5,34)( 6,36)( 7,32)( 8,31)( 9,33)(10,46)(11,48)(12,47)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,37)(20,39)(21,38)(22,44)(23,43)(24,45)(25,41)(26,40)(27,42)(55,58)(56,60)(57,59)(61,62)(64,76)(65,78)(66,77)(67,73)(68,75)(69,74)(70,80)(71,79)(72,81);
s3 := Sym(83)!(82,83);
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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s0*s1*s2*s0*s1*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;