# Polytope of Type {12,12}

Atlas Canonical Name : {12,12}*864l
if this polytope has a name.
Group : SmallGroup(864,4321)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 36, 216, 36
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{12,12,2} of size 1728
Vertex Figure Of :
{2,12,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,12}*432i
3-fold quotients : {12,4}*288
4-fold quotients : {6,12}*216c
6-fold quotients : {6,4}*144
9-fold quotients : {4,12}*96a
12-fold quotients : {6,4}*72
18-fold quotients : {2,12}*48, {4,6}*48a
27-fold quotients : {4,4}*32
36-fold quotients : {2,6}*24
54-fold quotients : {2,4}*16, {4,2}*16
72-fold quotients : {2,3}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,12}*1728t, {12,24}*1728u, {24,12}*1728v, {24,12}*1728w, {12,24}*1728x
Permutation Representation (GAP) :
```s0 := (  1, 55)(  2, 56)(  3, 57)(  4, 61)(  5, 62)(  6, 63)(  7, 58)(  8, 59)
(  9, 60)( 10, 73)( 11, 74)( 12, 75)( 13, 79)( 14, 80)( 15, 81)( 16, 76)
( 17, 77)( 18, 78)( 19, 64)( 20, 65)( 21, 66)( 22, 70)( 23, 71)( 24, 72)
( 25, 67)( 26, 68)( 27, 69)( 28, 82)( 29, 83)( 30, 84)( 31, 88)( 32, 89)
( 33, 90)( 34, 85)( 35, 86)( 36, 87)( 37,100)( 38,101)( 39,102)( 40,106)
( 41,107)( 42,108)( 43,103)( 44,104)( 45,105)( 46, 91)( 47, 92)( 48, 93)
( 49, 97)( 50, 98)( 51, 99)( 52, 94)( 53, 95)( 54, 96);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 16)( 11, 18)( 12, 17)( 14, 15)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 31)( 29, 33)( 30, 32)( 35, 36)
( 37, 43)( 38, 45)( 39, 44)( 41, 42)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 55, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)( 60, 83)( 61, 88)( 62, 90)
( 63, 89)( 64, 97)( 65, 99)( 66, 98)( 67, 94)( 68, 96)( 69, 95)( 70, 91)
( 71, 93)( 72, 92)( 73,100)( 74,102)( 75,101)( 76,106)( 77,108)( 78,107)
( 79,103)( 80,105)( 81,104);;
s2 := (  1,  2)(  4, 23)(  5, 22)(  6, 24)(  7, 17)(  8, 16)(  9, 18)( 10, 20)
( 11, 19)( 12, 21)( 13, 14)( 25, 26)( 28, 29)( 31, 50)( 32, 49)( 33, 51)
( 34, 44)( 35, 43)( 36, 45)( 37, 47)( 38, 46)( 39, 48)( 40, 41)( 52, 53)
( 55, 56)( 58, 77)( 59, 76)( 60, 78)( 61, 71)( 62, 70)( 63, 72)( 64, 74)
( 65, 73)( 66, 75)( 67, 68)( 79, 80)( 82, 83)( 85,104)( 86,103)( 87,105)
( 88, 98)( 89, 97)( 90, 99)( 91,101)( 92,100)( 93,102)( 94, 95)(106,107);;
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, s2*s0*s1*s2*s0*s1*s0*s1*s2*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*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*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, 55)(  2, 56)(  3, 57)(  4, 61)(  5, 62)(  6, 63)(  7, 58)
(  8, 59)(  9, 60)( 10, 73)( 11, 74)( 12, 75)( 13, 79)( 14, 80)( 15, 81)
( 16, 76)( 17, 77)( 18, 78)( 19, 64)( 20, 65)( 21, 66)( 22, 70)( 23, 71)
( 24, 72)( 25, 67)( 26, 68)( 27, 69)( 28, 82)( 29, 83)( 30, 84)( 31, 88)
( 32, 89)( 33, 90)( 34, 85)( 35, 86)( 36, 87)( 37,100)( 38,101)( 39,102)
( 40,106)( 41,107)( 42,108)( 43,103)( 44,104)( 45,105)( 46, 91)( 47, 92)
( 48, 93)( 49, 97)( 50, 98)( 51, 99)( 52, 94)( 53, 95)( 54, 96);
s1 := Sym(108)!(  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 16)( 11, 18)( 12, 17)
( 14, 15)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 31)( 29, 33)( 30, 32)
( 35, 36)( 37, 43)( 38, 45)( 39, 44)( 41, 42)( 47, 48)( 49, 52)( 50, 54)
( 51, 53)( 55, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)( 60, 83)( 61, 88)
( 62, 90)( 63, 89)( 64, 97)( 65, 99)( 66, 98)( 67, 94)( 68, 96)( 69, 95)
( 70, 91)( 71, 93)( 72, 92)( 73,100)( 74,102)( 75,101)( 76,106)( 77,108)
( 78,107)( 79,103)( 80,105)( 81,104);
s2 := Sym(108)!(  1,  2)(  4, 23)(  5, 22)(  6, 24)(  7, 17)(  8, 16)(  9, 18)
( 10, 20)( 11, 19)( 12, 21)( 13, 14)( 25, 26)( 28, 29)( 31, 50)( 32, 49)
( 33, 51)( 34, 44)( 35, 43)( 36, 45)( 37, 47)( 38, 46)( 39, 48)( 40, 41)
( 52, 53)( 55, 56)( 58, 77)( 59, 76)( 60, 78)( 61, 71)( 62, 70)( 63, 72)
( 64, 74)( 65, 73)( 66, 75)( 67, 68)( 79, 80)( 82, 83)( 85,104)( 86,103)
( 87,105)( 88, 98)( 89, 97)( 90, 99)( 91,101)( 92,100)( 93,102)( 94, 95)
(106,107);
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, s2*s0*s1*s2*s0*s1*s0*s1*s2*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*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*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