Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,3,6}

Atlas Canonical Name {4,3,6}*288

Overview

Group
SmallGroup(288,1028)
Rank
4
Schläfli Type
{4,3,6}
Vertices, edges, …
8, 12, 18, 6
Order of s0s1s2s3
6
Order of s0s1s2s3s2s1
2
Also known as
Dual of 4T4(1,1). if this polytope has another name.

Special Properties

  • Universal
  • Locally Toroidal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

12-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s1*s0*s1*s2)^2> of order 2

6 facets

  • 6 of 2-fold non-regular quotient of {4,3}*48

4 vertex figures

Representations

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

References

  1. Theorem 11C7,11C8, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambridge University Press, 2002)

to this polytope.