Part of the Atlas of Small Regular Polytopes

Polytope of Type {36,3}

Atlas Canonical Name {36,3}*1296

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,1784)
Rank
3
Schläfli Type
{36,3}
Vertices, edges, …
216, 324, 18
Order of s0s1s2
6
Order of s0s1s2s1
36
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • 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=<s0*s1*s2*(s1*s0)^7*s2*(s1*s0)^4*s1*s2> of order 2

12 facets

108 vertex figures

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

9 facets

108 vertex figures

P/N, where N=<(s0*s1)^4*s0*s2*(s1*s0)^3*s1*s2> of order 3

6 facets

72 vertex figures

P/N, where N=<(s0*s1)^4*s0*s2*(s1*s0)^3*s1*s2, s0*s1*s0*s2*(s1*s0)^8*s2*s1> of order 6

4 facets

36 vertex figures

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

3 facets

36 vertex figures

P/N, where N=<s0*s1*s2*(s1*s0)^3*s1*s2*s1> of order 6

4 facets

36 vertex figures

Representations

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

References

None.

to this polytope.

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