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# Polytope of Type {6,12,6}

Atlas Canonical Name : {6,12,6}*864e
if this polytope has a name.
Group : SmallGroup(864,4368)
Rank : 4
Schlafli Type : {6,12,6}
Number of vertices, edges, etc : 6, 36, 36, 6
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,12,6,2} of size 1728
Vertex Figure Of :
{2,6,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6,6}*432f
3-fold quotients : {2,12,6}*288b, {6,12,2}*288b
4-fold quotients : {6,3,6}*216
6-fold quotients : {2,6,6}*144c, {6,6,2}*144b
9-fold quotients : {2,12,2}*96
12-fold quotients : {2,3,6}*72, {6,3,2}*72
18-fold quotients : {2,6,2}*48
27-fold quotients : {2,4,2}*32
36-fold quotients : {2,3,2}*24
54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,24,6}*1728e, {6,12,12}*1728f, {12,12,6}*1728e
Permutation Representation (GAP) :
```s0 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 47, 48)
( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 65, 66)( 68, 69)( 71, 72)
( 74, 75)( 77, 78)( 80, 81)( 83, 84)( 86, 87)( 89, 90)( 92, 93)( 95, 96)
( 98, 99)(101,102)(104,105)(107,108);;
s1 := (  1,  2)(  4,  8)(  5,  7)(  6,  9)( 10, 20)( 11, 19)( 12, 21)( 13, 26)
( 14, 25)( 15, 27)( 16, 23)( 17, 22)( 18, 24)( 28, 29)( 31, 35)( 32, 34)
( 33, 36)( 37, 47)( 38, 46)( 39, 48)( 40, 53)( 41, 52)( 42, 54)( 43, 50)
( 44, 49)( 45, 51)( 55, 83)( 56, 82)( 57, 84)( 58, 89)( 59, 88)( 60, 90)
( 61, 86)( 62, 85)( 63, 87)( 64,101)( 65,100)( 66,102)( 67,107)( 68,106)
( 69,108)( 70,104)( 71,103)( 72,105)( 73, 92)( 74, 91)( 75, 93)( 76, 98)
( 77, 97)( 78, 99)( 79, 95)( 80, 94)( 81, 96);;
s2 := (  1, 67)(  2, 69)(  3, 68)(  4, 64)(  5, 66)(  6, 65)(  7, 70)(  8, 72)
(  9, 71)( 10, 58)( 11, 60)( 12, 59)( 13, 55)( 14, 57)( 15, 56)( 16, 61)
( 17, 63)( 18, 62)( 19, 76)( 20, 78)( 21, 77)( 22, 73)( 23, 75)( 24, 74)
( 25, 79)( 26, 81)( 27, 80)( 28, 94)( 29, 96)( 30, 95)( 31, 91)( 32, 93)
( 33, 92)( 34, 97)( 35, 99)( 36, 98)( 37, 85)( 38, 87)( 39, 86)( 40, 82)
( 41, 84)( 42, 83)( 43, 88)( 44, 90)( 45, 89)( 46,103)( 47,105)( 48,104)
( 49,100)( 50,102)( 51,101)( 52,106)( 53,108)( 54,107);;
s3 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108);;
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
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)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)
( 47, 48)( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 65, 66)( 68, 69)
( 71, 72)( 74, 75)( 77, 78)( 80, 81)( 83, 84)( 86, 87)( 89, 90)( 92, 93)
( 95, 96)( 98, 99)(101,102)(104,105)(107,108);
s1 := Sym(108)!(  1,  2)(  4,  8)(  5,  7)(  6,  9)( 10, 20)( 11, 19)( 12, 21)
( 13, 26)( 14, 25)( 15, 27)( 16, 23)( 17, 22)( 18, 24)( 28, 29)( 31, 35)
( 32, 34)( 33, 36)( 37, 47)( 38, 46)( 39, 48)( 40, 53)( 41, 52)( 42, 54)
( 43, 50)( 44, 49)( 45, 51)( 55, 83)( 56, 82)( 57, 84)( 58, 89)( 59, 88)
( 60, 90)( 61, 86)( 62, 85)( 63, 87)( 64,101)( 65,100)( 66,102)( 67,107)
( 68,106)( 69,108)( 70,104)( 71,103)( 72,105)( 73, 92)( 74, 91)( 75, 93)
( 76, 98)( 77, 97)( 78, 99)( 79, 95)( 80, 94)( 81, 96);
s2 := Sym(108)!(  1, 67)(  2, 69)(  3, 68)(  4, 64)(  5, 66)(  6, 65)(  7, 70)
(  8, 72)(  9, 71)( 10, 58)( 11, 60)( 12, 59)( 13, 55)( 14, 57)( 15, 56)
( 16, 61)( 17, 63)( 18, 62)( 19, 76)( 20, 78)( 21, 77)( 22, 73)( 23, 75)
( 24, 74)( 25, 79)( 26, 81)( 27, 80)( 28, 94)( 29, 96)( 30, 95)( 31, 91)
( 32, 93)( 33, 92)( 34, 97)( 35, 99)( 36, 98)( 37, 85)( 38, 87)( 39, 86)
( 40, 82)( 41, 84)( 42, 83)( 43, 88)( 44, 90)( 45, 89)( 46,103)( 47,105)
( 48,104)( 49,100)( 50,102)( 51,101)( 52,106)( 53,108)( 54,107);
s3 := Sym(108)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)
( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108);
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,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
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