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# Polytope of Type {2,12,12}

Atlas Canonical Name : {2,12,12}*1728f
if this polytope has a name.
Group : SmallGroup(1728,30413)
Rank : 4
Schlafli Type : {2,12,12}
Number of vertices, edges, etc : 2, 36, 216, 36
Order of s0s1s2s3 : 4
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,6,12}*864f
3-fold quotients : {2,12,4}*576
4-fold quotients : {2,6,12}*432b
6-fold quotients : {2,6,4}*288
12-fold quotients : {2,6,4}*144
27-fold quotients : {2,4,4}*64
54-fold quotients : {2,2,4}*32, {2,4,2}*32
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (  3, 57)(  4, 58)(  5, 59)(  6, 65)(  7, 63)(  8, 64)(  9, 61)( 10, 62)
( 11, 60)( 12, 75)( 13, 76)( 14, 77)( 15, 83)( 16, 81)( 17, 82)( 18, 79)
( 19, 80)( 20, 78)( 21, 66)( 22, 67)( 23, 68)( 24, 74)( 25, 72)( 26, 73)
( 27, 70)( 28, 71)( 29, 69)( 30, 84)( 31, 85)( 32, 86)( 33, 92)( 34, 90)
( 35, 91)( 36, 88)( 37, 89)( 38, 87)( 39,102)( 40,103)( 41,104)( 42,110)
( 43,108)( 44,109)( 45,106)( 46,107)( 47,105)( 48, 93)( 49, 94)( 50, 95)
( 51,101)( 52, 99)( 53,100)( 54, 97)( 55, 98)( 56, 96);;
s2 := (  3, 15)(  4, 17)(  5, 16)(  6,  8)(  9, 25)( 10, 24)( 11, 26)( 13, 14)
( 18, 21)( 19, 23)( 20, 22)( 27, 29)( 30, 42)( 31, 44)( 32, 43)( 33, 35)
( 36, 52)( 37, 51)( 38, 53)( 40, 41)( 45, 48)( 46, 50)( 47, 49)( 54, 56)
( 57, 96)( 58, 98)( 59, 97)( 60, 89)( 61, 88)( 62, 87)( 63,106)( 64,105)
( 65,107)( 66, 93)( 67, 95)( 68, 94)( 69, 84)( 70, 86)( 71, 85)( 72,102)
( 73,104)( 74,103)( 75, 99)( 76,101)( 77,100)( 78, 91)( 79, 90)( 80, 92)
( 81,110)( 82,109)( 83,108);;
s3 := (  4,  5)(  6,  7)(  9, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 25)( 16, 24)
( 17, 26)( 18, 29)( 19, 28)( 20, 27)( 31, 32)( 33, 34)( 36, 38)( 39, 48)
( 40, 50)( 41, 49)( 42, 52)( 43, 51)( 44, 53)( 45, 56)( 46, 55)( 47, 54)
( 58, 59)( 60, 61)( 63, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 79)( 70, 78)
( 71, 80)( 72, 83)( 73, 82)( 74, 81)( 85, 86)( 87, 88)( 90, 92)( 93,102)
( 94,104)( 95,103)( 96,106)( 97,105)( 98,107)( 99,110)(100,109)(101,108);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s1*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s3*s2,
s3*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*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*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```

to this polytope