Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,2,4,9}

Atlas Canonical Name {6,2,4,9}*1728

Overview

Group
SmallGroup(1728,46115)
Rank
5
Schläfli Type
{6,2,4,9}
Vertices, edges, …
6, 6, 8, 36, 18
Order of s0s1s2s3s4
18
Order of s0s1s2s3s4s3s2s1
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

None in this atlas.

Representations

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