Part of the Atlas of Small Regular Polytopes

Polytope of Type {72,2,3}

Atlas Canonical Name {72,2,3}*864

Overview

Group
SmallGroup(864,770)
Rank
4
Schläfli Type
{72,2,3}
Vertices, edges, …
72, 72, 3, 3
Order of s0s1s2s3
72
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

8-fold

9-fold

12-fold

18-fold

24-fold

36-fold

Covers minimal covers in bold

2-fold

Representations

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