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# Polytope of Type {2,12,6}

Atlas Canonical Name : {2,12,6}*864c
if this polytope has a name.
Group : SmallGroup(864,2511)
Rank : 4
Schlafli Type : {2,12,6}
Number of vertices, edges, etc : 2, 36, 108, 18
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,12,6,2} of size 1728
Vertex Figure Of :
{2,2,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,6,6}*432a
3-fold quotients : {2,12,6}*288c
4-fold quotients : {2,6,3}*216
6-fold quotients : {2,6,6}*144b
9-fold quotients : {2,4,6}*96a
12-fold quotients : {2,6,3}*72
18-fold quotients : {2,2,6}*48
27-fold quotients : {2,4,2}*32
36-fold quotients : {2,2,3}*24
54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,12,12}*1728b, {2,24,6}*1728c, {4,12,6}*1728c
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)
( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)
( 57, 84)( 58, 86)( 59, 85)( 60, 90)( 61, 92)( 62, 91)( 63, 87)( 64, 89)
( 65, 88)( 66, 93)( 67, 95)( 68, 94)( 69, 99)( 70,101)( 71,100)( 72, 96)
( 73, 98)( 74, 97)( 75,102)( 76,104)( 77,103)( 78,108)( 79,110)( 80,109)
( 81,105)( 82,107)( 83,106);;
s2 := (  3, 60)(  4, 61)(  5, 62)(  6, 57)(  7, 58)(  8, 59)(  9, 63)( 10, 64)
( 11, 65)( 12, 78)( 13, 79)( 14, 80)( 15, 75)( 16, 76)( 17, 77)( 18, 81)
( 19, 82)( 20, 83)( 21, 69)( 22, 70)( 23, 71)( 24, 66)( 25, 67)( 26, 68)
( 27, 72)( 28, 73)( 29, 74)( 30, 87)( 31, 88)( 32, 89)( 33, 84)( 34, 85)
( 35, 86)( 36, 90)( 37, 91)( 38, 92)( 39,105)( 40,106)( 41,107)( 42,102)
( 43,103)( 44,104)( 45,108)( 46,109)( 47,110)( 48, 96)( 49, 97)( 50, 98)
( 51, 93)( 52, 94)( 53, 95)( 54, 99)( 55,100)( 56,101);;
s3 := (  3, 12)(  4, 13)(  5, 14)(  6, 20)(  7, 18)(  8, 19)(  9, 16)( 10, 17)
( 11, 15)( 24, 29)( 25, 27)( 26, 28)( 30, 39)( 31, 40)( 32, 41)( 33, 47)
( 34, 45)( 35, 46)( 36, 43)( 37, 44)( 38, 42)( 51, 56)( 52, 54)( 53, 55)
( 57, 66)( 58, 67)( 59, 68)( 60, 74)( 61, 72)( 62, 73)( 63, 70)( 64, 71)
( 65, 69)( 78, 83)( 79, 81)( 80, 82)( 84, 93)( 85, 94)( 86, 95)( 87,101)
( 88, 99)( 89,100)( 90, 97)( 91, 98)( 92, 96)(105,110)(106,108)(107,109);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(110)!(1,2);
s1 := Sym(110)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)
( 53, 55)( 57, 84)( 58, 86)( 59, 85)( 60, 90)( 61, 92)( 62, 91)( 63, 87)
( 64, 89)( 65, 88)( 66, 93)( 67, 95)( 68, 94)( 69, 99)( 70,101)( 71,100)
( 72, 96)( 73, 98)( 74, 97)( 75,102)( 76,104)( 77,103)( 78,108)( 79,110)
( 80,109)( 81,105)( 82,107)( 83,106);
s2 := Sym(110)!(  3, 60)(  4, 61)(  5, 62)(  6, 57)(  7, 58)(  8, 59)(  9, 63)
( 10, 64)( 11, 65)( 12, 78)( 13, 79)( 14, 80)( 15, 75)( 16, 76)( 17, 77)
( 18, 81)( 19, 82)( 20, 83)( 21, 69)( 22, 70)( 23, 71)( 24, 66)( 25, 67)
( 26, 68)( 27, 72)( 28, 73)( 29, 74)( 30, 87)( 31, 88)( 32, 89)( 33, 84)
( 34, 85)( 35, 86)( 36, 90)( 37, 91)( 38, 92)( 39,105)( 40,106)( 41,107)
( 42,102)( 43,103)( 44,104)( 45,108)( 46,109)( 47,110)( 48, 96)( 49, 97)
( 50, 98)( 51, 93)( 52, 94)( 53, 95)( 54, 99)( 55,100)( 56,101);
s3 := Sym(110)!(  3, 12)(  4, 13)(  5, 14)(  6, 20)(  7, 18)(  8, 19)(  9, 16)
( 10, 17)( 11, 15)( 24, 29)( 25, 27)( 26, 28)( 30, 39)( 31, 40)( 32, 41)
( 33, 47)( 34, 45)( 35, 46)( 36, 43)( 37, 44)( 38, 42)( 51, 56)( 52, 54)
( 53, 55)( 57, 66)( 58, 67)( 59, 68)( 60, 74)( 61, 72)( 62, 73)( 63, 70)
( 64, 71)( 65, 69)( 78, 83)( 79, 81)( 80, 82)( 84, 93)( 85, 94)( 86, 95)
( 87,101)( 88, 99)( 89,100)( 90, 97)( 91, 98)( 92, 96)(105,110)(106,108)
(107,109);
poly := sub<Sym(110)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2 >;

```

to this polytope