Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,12}

Atlas Canonical Name {4,12}*864b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(864,2282)
Rank
3
Schläfli Type
{4,12}
Vertices, edges, …
36, 216, 108
Order of s0s1s2
12
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

12-fold

27-fold

54-fold

108-fold

Covers minimal covers in bold

2-fold

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)^2*(s2*s1*s0*s1)^2*s2> of order 2

54 facets

18 vertex figures

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

60 facets

18 vertex figures

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

36 facets

12 vertex figures

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

36 facets

16 vertex figures

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

24 facets

6 vertex figures

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

24 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

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