Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6}

Atlas Canonical Name {3,6}*768

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Overview

Group
SmallGroup(768,1085833)
Rank
3
Schläfli Type
{3,6}
Vertices, edges, …
64, 192, 128
Order of s0s1s2
16
Order of s0s1s2s1
6
Also known as
{3,6}(8,0), {3,6}16. if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

4-fold

16-fold

32-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s0*s1*(s2*s1*s0)^6*(s2*s1)^2> of order 2

64 facets

32 vertex figures

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

64 facets

34 vertex figures

P/N, where N=<s1*(s0*(s2*s1)^2)^2*s0*(s2*s1)^2*s0*s2*s1*s2> of order 4

32 facets

16 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^2*(s2*s1)^2> of order 4

32 facets

16 vertex figures

P/N, where N=<s0*s2*s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*(s2*s1*s0)^6*(s2*s1)^2> of order 4

32 facets

18 vertex figures

P/N, where N=<(s1*s0*s2)^7*s1*s2> of order 4

32 facets

16 vertex figures

P/N, where N=<s0*s1*s0*s2*s1*(s0*(s2*s1)^2)^2> of order 4

32 facets

16 vertex figures

P/N, where N=<s0*s2*s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*s0*s2*s1*(s0*(s2*s1)^2)^2> of order 8

16 facets

9 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^2*(s2*s1)^2, s0*s2*s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 8

16 facets

10 vertex figures

P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s2*s1, s0*s2*s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 8

16 facets

10 vertex figures

P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 8

16 facets

8 vertex figures

P/N, where N=<s0*s1*s0*s2*s1*(s0*(s2*s1)^2)^2*s2> of order 8

16 facets

8 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^2*(s2*s1)^2, s0*s1*(s0*(s2*s1)^2)^2*s0*(s2*s1)^2*s0*s2*s1*s2> of order 8

16 facets

8 vertex figures

P/N, where N=<s0*s1*s0*s2*s1*(s0*(s2*s1)^2)^2, (s0*s2*s1)^2*(s0*(s2*s1)^2)^2*s2> of order 8

16 facets

8 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^2*(s2*s1)^2, s0*(s1*s0*s2)^2*s1*(s0*(s2*s1)^2)^2*s0*s2*s1*s2> of order 8

16 facets

8 vertex figures

P/N, where N=<(s1*s0*s2)^3*s1*s2> of order 8

16 facets

8 vertex figures

P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s2*s1, s0*s2*s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*(s2*s1*s0)^2*(s2*s1)^2*s2> of order 16

8 facets

5 vertex figures

P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*(s2*s1*s0)^2*(s2*s1)^2*s2> of order 16

8 facets

4 vertex figures

Representations

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

References

None.

to this polytope.

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