Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,2,6,27}

Atlas Canonical Name {2,2,6,27}*1296

Overview

Group
SmallGroup(1296,1859)
Rank
5
Schläfli Type
{2,2,6,27}
Vertices, edges, …
2, 2, 6, 81, 27
Order of s0s1s2s3s4
54
Order of s0s1s2s3s4s3s2s1
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

None in this atlas.

Representations

Permutation Representation (GAP) 
s0 := (1,2);;
s1 := (3,4);;
s2 := (14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)(49,58)(68,77)(69,78)(70,79)(71,80)(72,81)(73,82)(74,83)(75,84)(76,85);;
s3 := ( 5,14)( 6,16)( 7,15)( 8,21)( 9,20)(10,22)(11,18)(12,17)(13,19)(24,25)(26,30)(27,29)(28,31)(32,71)(33,73)(34,72)(35,68)(36,70)(37,69)(38,75)(39,74)(40,76)(41,62)(42,64)(43,63)(44,59)(45,61)(46,60)(47,66)(48,65)(49,67)(50,80)(51,82)(52,81)(53,77)(54,79)(55,78)(56,84)(57,83)(58,85);;
s4 := ( 5,32)( 6,34)( 7,33)( 8,39)( 9,38)(10,40)(11,36)(12,35)(13,37)(14,50)(15,52)(16,51)(17,57)(18,56)(19,58)(20,54)(21,53)(22,55)(23,41)(24,43)(25,42)(26,48)(27,47)(28,49)(29,45)(30,44)(31,46)(59,62)(60,64)(61,63)(65,66)(68,80)(69,82)(70,81)(71,77)(72,79)(73,78)(74,84)(75,83)(76,85);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s2*s3*s4*s2*s3*s2*s3*s4*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(85)!(1,2);
s1 := Sym(85)!(3,4);
s2 := Sym(85)!(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)(49,58)(68,77)(69,78)(70,79)(71,80)(72,81)(73,82)(74,83)(75,84)(76,85);
s3 := Sym(85)!( 5,14)( 6,16)( 7,15)( 8,21)( 9,20)(10,22)(11,18)(12,17)(13,19)(24,25)(26,30)(27,29)(28,31)(32,71)(33,73)(34,72)(35,68)(36,70)(37,69)(38,75)(39,74)(40,76)(41,62)(42,64)(43,63)(44,59)(45,61)(46,60)(47,66)(48,65)(49,67)(50,80)(51,82)(52,81)(53,77)(54,79)(55,78)(56,84)(57,83)(58,85);
s4 := Sym(85)!( 5,32)( 6,34)( 7,33)( 8,39)( 9,38)(10,40)(11,36)(12,35)(13,37)(14,50)(15,52)(16,51)(17,57)(18,56)(19,58)(20,54)(21,53)(22,55)(23,41)(24,43)(25,42)(26,48)(27,47)(28,49)(29,45)(30,44)(31,46)(59,62)(60,64)(61,63)(65,66)(68,80)(69,82)(70,81)(71,77)(72,79)(73,78)(74,84)(75,83)(76,85);
poly := sub<Sym(85)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s2*s3*s4*s2*s3*s2*s3*s4*s2*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;