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# Polytope of Type {12,4}

Atlas Canonical Name : {12,4}*864a
if this polytope has a name.
Group : SmallGroup(864,2282)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 108, 216, 36
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{12,4,2} of size 1728
Vertex Figure Of :
{2,12,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,4}*432a
3-fold quotients : {4,4}*288
4-fold quotients : {12,4}*216
6-fold quotients : {4,4}*144
12-fold quotients : {4,4}*72
27-fold quotients : {4,4}*32
54-fold quotients : {2,4}*16, {4,2}*16
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,4}*1728a, {12,8}*1728b, {24,4}*1728b, {12,8}*1728c, {24,4}*1728d
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 := (  1,  2)(  4, 19)(  5, 21)(  6, 20)(  7, 12)(  8, 11)(  9, 10)( 13, 27)
( 14, 26)( 15, 25)( 16, 18)( 22, 23)( 28, 29)( 31, 46)( 32, 48)( 33, 47)
( 34, 39)( 35, 38)( 36, 37)( 40, 54)( 41, 53)( 42, 52)( 43, 45)( 49, 50)
( 55, 83)( 56, 82)( 57, 84)( 58,100)( 59,102)( 60,101)( 61, 93)( 62, 92)
( 63, 91)( 64, 90)( 65, 89)( 66, 88)( 67,108)( 68,107)( 69,106)( 70, 99)
( 71, 98)( 72, 97)( 73, 85)( 74, 87)( 75, 86)( 76,104)( 77,103)( 78,105)
( 79, 96)( 80, 95)( 81, 94);;
s2 := (  1, 62)(  2, 61)(  3, 63)(  4, 59)(  5, 58)(  6, 60)(  7, 56)(  8, 55)
(  9, 57)( 10, 71)( 11, 70)( 12, 72)( 13, 68)( 14, 67)( 15, 69)( 16, 65)
( 17, 64)( 18, 66)( 19, 80)( 20, 79)( 21, 81)( 22, 77)( 23, 76)( 24, 78)
( 25, 74)( 26, 73)( 27, 75)( 28, 89)( 29, 88)( 30, 90)( 31, 86)( 32, 85)
( 33, 87)( 34, 83)( 35, 82)( 36, 84)( 37, 98)( 38, 97)( 39, 99)( 40, 95)
( 41, 94)( 42, 96)( 43, 92)( 44, 91)( 45, 93)( 46,107)( 47,106)( 48,108)
( 49,104)( 50,103)( 51,105)( 52,101)( 53,100)( 54,102);;
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,
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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)!(  1,  2)(  4, 19)(  5, 21)(  6, 20)(  7, 12)(  8, 11)(  9, 10)
( 13, 27)( 14, 26)( 15, 25)( 16, 18)( 22, 23)( 28, 29)( 31, 46)( 32, 48)
( 33, 47)( 34, 39)( 35, 38)( 36, 37)( 40, 54)( 41, 53)( 42, 52)( 43, 45)
( 49, 50)( 55, 83)( 56, 82)( 57, 84)( 58,100)( 59,102)( 60,101)( 61, 93)
( 62, 92)( 63, 91)( 64, 90)( 65, 89)( 66, 88)( 67,108)( 68,107)( 69,106)
( 70, 99)( 71, 98)( 72, 97)( 73, 85)( 74, 87)( 75, 86)( 76,104)( 77,103)
( 78,105)( 79, 96)( 80, 95)( 81, 94);
s2 := Sym(108)!(  1, 62)(  2, 61)(  3, 63)(  4, 59)(  5, 58)(  6, 60)(  7, 56)
(  8, 55)(  9, 57)( 10, 71)( 11, 70)( 12, 72)( 13, 68)( 14, 67)( 15, 69)
( 16, 65)( 17, 64)( 18, 66)( 19, 80)( 20, 79)( 21, 81)( 22, 77)( 23, 76)
( 24, 78)( 25, 74)( 26, 73)( 27, 75)( 28, 89)( 29, 88)( 30, 90)( 31, 86)
( 32, 85)( 33, 87)( 34, 83)( 35, 82)( 36, 84)( 37, 98)( 38, 97)( 39, 99)
( 40, 95)( 41, 94)( 42, 96)( 43, 92)( 44, 91)( 45, 93)( 46,107)( 47,106)
( 48,108)( 49,104)( 50,103)( 51,105)( 52,101)( 53,100)( 54,102);
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,
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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