Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,2,18,4}

Atlas Canonical Name {6,2,18,4}*1728c

Overview

Group
SmallGroup(1728,46115)
Rank
5
Schläfli Type
{6,2,18,4}
Vertices, edges, …
6, 6, 18, 36, 4
Order of s0s1s2s3s4
18
Order of s0s1s2s3s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

9-fold

12-fold

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