Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6}

Atlas Canonical Name {3,6}*1296

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Overview

Group
SmallGroup(1296,1784)
Rank
3
Schläfli Type
{3,6}
Vertices, edges, …
108, 324, 216
Order of s0s1s2
36
Order of s0s1s2s1
6
Also known as
{3,6}(6,6). if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

4-fold

9-fold

12-fold

27-fold

36-fold

54-fold

108-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=<s1*((s0*(s2*s1)^2)^2)^2*s0*(s2*s1)^2*s0*s2*s1*s2> of order 2

108 facets

54 vertex figures

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

108 facets

56 vertex figures

P/N, where N=<((s0*(s2*s1)^2)^2)^2> of order 3

72 facets

36 vertex figures

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

72 facets

38 vertex figures

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

54 facets

27 vertex figures

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

54 facets

27 vertex figures

P/N, where N=<((s0*(s2*s1)^2)^2)^2, s0*(s1*s0*s2)^5*s1*(s0*(s2*s1)^2)^2*s2> of order 6

36 facets

18 vertex figures

P/N, where N=<((s0*(s2*s1)^2)^2)^2, s0*(s1*s0*s2)^5*s1*(s0*(s2*s1)^2)^2*s0> of order 6

36 facets

20 vertex figures

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

36 facets

18 vertex figures

P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*(s2*s1*s0)^7*(s2*s1)^2> of order 12

18 facets

9 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 21)( 14, 22)( 15, 24)( 16, 23)( 19, 20)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 35, 36)( 37, 77)( 38, 78)( 39, 80)( 40, 79)( 41, 73)( 42, 74)( 43, 76)( 44, 75)( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 85)( 50, 86)( 51, 88)( 52, 87)( 53, 93)( 54, 94)( 55, 96)( 56, 95)( 57, 89)( 58, 90)( 59, 92)( 60, 91)( 61,105)( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)( 67,104)( 68,103)( 69, 97)( 70, 98)( 71,100)( 72, 99);;
s1 := (  1, 49)(  2, 51)(  3, 50)(  4, 52)(  5, 57)(  6, 59)(  7, 58)(  8, 60)(  9, 53)( 10, 55)( 11, 54)( 12, 56)( 13, 65)( 14, 67)( 15, 66)( 16, 68)( 17, 61)( 18, 63)( 19, 62)( 20, 64)( 21, 69)( 22, 71)( 23, 70)( 24, 72)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 41)( 30, 43)( 31, 42)( 32, 44)( 33, 37)( 34, 39)( 35, 38)( 36, 40)( 73, 81)( 74, 83)( 75, 82)( 76, 84)( 78, 79)( 86, 87)( 89, 93)( 90, 95)( 91, 94)( 92, 96)( 97,101)( 98,103)( 99,102)(100,104)(106,107);;
s2 := (  1,  2)(  5,  6)(  9, 10)( 13, 30)( 14, 29)( 15, 31)( 16, 32)( 17, 34)( 18, 33)( 19, 35)( 20, 36)( 21, 26)( 22, 25)( 23, 27)( 24, 28)( 37, 38)( 41, 42)( 45, 46)( 49, 66)( 50, 65)( 51, 67)( 52, 68)( 53, 70)( 54, 69)( 55, 71)( 56, 72)( 57, 62)( 58, 61)( 59, 63)( 60, 64)( 73, 74)( 77, 78)( 81, 82)( 85,102)( 86,101)( 87,103)( 88,104)( 89,106)( 90,105)( 91,107)( 92,108)( 93, 98)( 94, 97)( 95, 99)( 96,100);;
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*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(108)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 21)( 14, 22)( 15, 24)( 16, 23)( 19, 20)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 35, 36)( 37, 77)( 38, 78)( 39, 80)( 40, 79)( 41, 73)( 42, 74)( 43, 76)( 44, 75)( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 85)( 50, 86)( 51, 88)( 52, 87)( 53, 93)( 54, 94)( 55, 96)( 56, 95)( 57, 89)( 58, 90)( 59, 92)( 60, 91)( 61,105)( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)( 67,104)( 68,103)( 69, 97)( 70, 98)( 71,100)( 72, 99);
s1 := Sym(108)!(  1, 49)(  2, 51)(  3, 50)(  4, 52)(  5, 57)(  6, 59)(  7, 58)(  8, 60)(  9, 53)( 10, 55)( 11, 54)( 12, 56)( 13, 65)( 14, 67)( 15, 66)( 16, 68)( 17, 61)( 18, 63)( 19, 62)( 20, 64)( 21, 69)( 22, 71)( 23, 70)( 24, 72)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 41)( 30, 43)( 31, 42)( 32, 44)( 33, 37)( 34, 39)( 35, 38)( 36, 40)( 73, 81)( 74, 83)( 75, 82)( 76, 84)( 78, 79)( 86, 87)( 89, 93)( 90, 95)( 91, 94)( 92, 96)( 97,101)( 98,103)( 99,102)(100,104)(106,107);
s2 := Sym(108)!(  1,  2)(  5,  6)(  9, 10)( 13, 30)( 14, 29)( 15, 31)( 16, 32)( 17, 34)( 18, 33)( 19, 35)( 20, 36)( 21, 26)( 22, 25)( 23, 27)( 24, 28)( 37, 38)( 41, 42)( 45, 46)( 49, 66)( 50, 65)( 51, 67)( 52, 68)( 53, 70)( 54, 69)( 55, 71)( 56, 72)( 57, 62)( 58, 61)( 59, 63)( 60, 64)( 73, 74)( 77, 78)( 81, 82)( 85,102)( 86,101)( 87,103)( 88,104)( 89,106)( 90,105)( 91,107)( 92,108)( 93, 98)( 94, 97)( 95, 99)( 96,100);
poly := sub<Sym(108)|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*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;

References

None.

to this polytope.

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