Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6}

Atlas Canonical Name {12,6}*1296f

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Overview

Group
SmallGroup(1296,1790)
Rank
3
Schläfli Type
{12,6}
Vertices, edges, …
108, 324, 54
Order of s0s1s2
9
Order of s0s1s2s1
12
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable

Quotients maximal quotients in bold

3-fold

4-fold

9-fold

12-fold

27-fold

54-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)^6> of order 2

36 facets

54 vertex figures

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

30 facets

36 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

24 facets

18 vertex figures

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

18 facets

18 vertex figures

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

12 facets

18 vertex figures

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

12 facets

18 vertex figures

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

12 facets

18 vertex figures

P/N, where N=<(s1*s0*(s1*s2)^2)^2, s1*s0*s2*(s1*s0)^3*s1*s2*s1> of order 9

10 facets

12 vertex figures

P/N, where N=<(s0*s1)^2*s2*s1*s0*(s2*s1)^2, s1*s0*(s2*s1)^2*(s0*s1)^2*s2> of order 9

6 facets

12 vertex figures

P/N, where N=<(s0*s1)^4, s0*s2*(s1*s0)^3*s1*s2> of order 9

14 facets

12 vertex figures

P/N, where N=<s0*s2*(s1*s0)^3*s1*s2, s0*(s2*s1)^2*s0*(s1*s2)^2> of order 18

8 facets

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

References

None.

to this polytope.

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