Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,9,6}

Atlas Canonical Name {4,9,6}*1296d

Overview

Group
SmallGroup(1296,1790)
Rank
4
Schläfli Type
{4,9,6}
Vertices, edges, …
4, 54, 81, 18
Order of s0s1s2s3
6
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

27-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=<s2*s1*s2*s3*s2*s1*s3*s2> of order 3

6 facets

4 vertex figures

P/N, where N=<s1*s2*s1*s3*s2*s1*(s2*s3)^3> of order 3

6 facets

4 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108);;
s1 := (  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, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 81)( 42, 82)( 43, 84)( 44, 83)( 45, 77)( 46, 78)( 47, 80)( 48, 79)( 49, 93)( 50, 94)( 51, 96)( 52, 95)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)( 60, 87)( 61,101)( 62,102)( 63,104)( 64,103)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69,105)( 70,106)( 71,108)( 72,107);;
s2 := (  1, 85)(  2, 88)(  3, 87)(  4, 86)(  5, 93)(  6, 96)(  7, 95)(  8, 94)(  9, 89)( 10, 92)( 11, 91)( 12, 90)( 13, 97)( 14,100)( 15, 99)( 16, 98)( 17,105)( 18,108)( 19,107)( 20,106)( 21,101)( 22,104)( 23,103)( 24,102)( 25, 73)( 26, 76)( 27, 75)( 28, 74)( 29, 81)( 30, 84)( 31, 83)( 32, 82)( 33, 77)( 34, 80)( 35, 79)( 36, 78)( 38, 40)( 41, 45)( 42, 48)( 43, 47)( 44, 46)( 50, 52)( 53, 57)( 54, 60)( 55, 59)( 56, 58)( 62, 64)( 65, 69)( 66, 72)( 67, 71)( 68, 70);;
s3 := (  5,  9)(  6, 10)(  7, 11)(  8, 12)( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 33)( 18, 34)( 19, 35)( 20, 36)( 21, 29)( 22, 30)( 23, 31)( 24, 32)( 37, 73)( 38, 74)( 39, 75)( 40, 76)( 41, 81)( 42, 82)( 43, 83)( 44, 84)( 45, 77)( 46, 78)( 47, 79)( 48, 80)( 49, 97)( 50, 98)( 51, 99)( 52,100)( 53,105)( 54,106)( 55,107)( 56,108)( 57,101)( 58,102)( 59,103)( 60,104)( 61, 85)( 62, 86)( 63, 87)( 64, 88)( 65, 93)( 66, 94)( 67, 95)( 68, 96)( 69, 89)( 70, 90)( 71, 91)( 72, 92);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(108)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108);
s1 := 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, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 81)( 42, 82)( 43, 84)( 44, 83)( 45, 77)( 46, 78)( 47, 80)( 48, 79)( 49, 93)( 50, 94)( 51, 96)( 52, 95)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)( 60, 87)( 61,101)( 62,102)( 63,104)( 64,103)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69,105)( 70,106)( 71,108)( 72,107);
s2 := Sym(108)!(  1, 85)(  2, 88)(  3, 87)(  4, 86)(  5, 93)(  6, 96)(  7, 95)(  8, 94)(  9, 89)( 10, 92)( 11, 91)( 12, 90)( 13, 97)( 14,100)( 15, 99)( 16, 98)( 17,105)( 18,108)( 19,107)( 20,106)( 21,101)( 22,104)( 23,103)( 24,102)( 25, 73)( 26, 76)( 27, 75)( 28, 74)( 29, 81)( 30, 84)( 31, 83)( 32, 82)( 33, 77)( 34, 80)( 35, 79)( 36, 78)( 38, 40)( 41, 45)( 42, 48)( 43, 47)( 44, 46)( 50, 52)( 53, 57)( 54, 60)( 55, 59)( 56, 58)( 62, 64)( 65, 69)( 66, 72)( 67, 71)( 68, 70);
s3 := Sym(108)!(  5,  9)(  6, 10)(  7, 11)(  8, 12)( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 33)( 18, 34)( 19, 35)( 20, 36)( 21, 29)( 22, 30)( 23, 31)( 24, 32)( 37, 73)( 38, 74)( 39, 75)( 40, 76)( 41, 81)( 42, 82)( 43, 83)( 44, 84)( 45, 77)( 46, 78)( 47, 79)( 48, 80)( 49, 97)( 50, 98)( 51, 99)( 52,100)( 53,105)( 54,106)( 55,107)( 56,108)( 57,101)( 58,102)( 59,103)( 60,104)( 61, 85)( 62, 86)( 63, 87)( 64, 88)( 65, 93)( 66, 94)( 67, 95)( 68, 96)( 69, 89)( 70, 90)( 71, 91)( 72, 92);
poly := sub<Sym(108)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References

None.

to this polytope.