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# Polytope of Type {2,6,12,6}

Atlas Canonical Name : {2,6,12,6}*1728f
if this polytope has a name.
Group : SmallGroup(1728,47409)
Rank : 5
Schlafli Type : {2,6,12,6}
Number of vertices, edges, etc : 2, 6, 36, 36, 6
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 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 : {2,6,6,6}*864g
3-fold quotients : {2,6,4,6}*576, {2,6,12,2}*576c
4-fold quotients : {2,3,6,6}*432b
6-fold quotients : {2,6,2,6}*288, {2,6,6,2}*288c
9-fold quotients : {2,2,4,6}*192a, {2,6,4,2}*192a
12-fold quotients : {2,3,2,6}*144, {2,3,6,2}*144, {2,6,2,3}*144
18-fold quotients : {2,2,2,6}*96, {2,6,2,2}*96
24-fold quotients : {2,3,2,3}*72
27-fold quotients : {2,2,4,2}*64
36-fold quotients : {2,2,2,3}*48, {2,3,2,2}*48
54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)( 16, 28)
( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 33, 36)( 34, 37)( 35, 38)( 39, 48)
( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)( 47, 53)
( 60, 63)( 61, 64)( 62, 65)( 66, 75)( 67, 76)( 68, 77)( 69, 81)( 70, 82)
( 71, 83)( 72, 78)( 73, 79)( 74, 80)( 87, 90)( 88, 91)( 89, 92)( 93,102)
( 94,103)( 95,104)( 96,108)( 97,109)( 98,110)( 99,105)(100,106)(101,107);;
s2 := (  3, 69)(  4, 70)(  5, 71)(  6, 66)(  7, 67)(  8, 68)(  9, 72)( 10, 73)
( 11, 74)( 12, 60)( 13, 61)( 14, 62)( 15, 57)( 16, 58)( 17, 59)( 18, 63)
( 19, 64)( 20, 65)( 21, 78)( 22, 79)( 23, 80)( 24, 75)( 25, 76)( 26, 77)
( 27, 81)( 28, 82)( 29, 83)( 30, 96)( 31, 97)( 32, 98)( 33, 93)( 34, 94)
( 35, 95)( 36, 99)( 37,100)( 38,101)( 39, 87)( 40, 88)( 41, 89)( 42, 84)
( 43, 85)( 44, 86)( 45, 90)( 46, 91)( 47, 92)( 48,105)( 49,106)( 50,107)
( 51,102)( 52,103)( 53,104)( 54,108)( 55,109)( 56,110);;
s3 := (  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);;
s4 := (  3,  4)(  6,  7)(  9, 10)( 12, 13)( 15, 16)( 18, 19)( 21, 22)( 24, 25)
( 27, 28)( 30, 31)( 33, 34)( 36, 37)( 39, 40)( 42, 43)( 45, 46)( 48, 49)
( 51, 52)( 54, 55)( 57, 58)( 60, 61)( 63, 64)( 66, 67)( 69, 70)( 72, 73)
( 75, 76)( 78, 79)( 81, 82)( 84, 85)( 87, 88)( 90, 91)( 93, 94)( 96, 97)
( 99,100)(102,103)(105,106)(108,109);;
poly := Group([s0,s1,s2,s3,s4]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(110)!(1,2);
s1 := Sym(110)!(  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)
( 16, 28)( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 33, 36)( 34, 37)( 35, 38)
( 39, 48)( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)
( 47, 53)( 60, 63)( 61, 64)( 62, 65)( 66, 75)( 67, 76)( 68, 77)( 69, 81)
( 70, 82)( 71, 83)( 72, 78)( 73, 79)( 74, 80)( 87, 90)( 88, 91)( 89, 92)
( 93,102)( 94,103)( 95,104)( 96,108)( 97,109)( 98,110)( 99,105)(100,106)
(101,107);
s2 := Sym(110)!(  3, 69)(  4, 70)(  5, 71)(  6, 66)(  7, 67)(  8, 68)(  9, 72)
( 10, 73)( 11, 74)( 12, 60)( 13, 61)( 14, 62)( 15, 57)( 16, 58)( 17, 59)
( 18, 63)( 19, 64)( 20, 65)( 21, 78)( 22, 79)( 23, 80)( 24, 75)( 25, 76)
( 26, 77)( 27, 81)( 28, 82)( 29, 83)( 30, 96)( 31, 97)( 32, 98)( 33, 93)
( 34, 94)( 35, 95)( 36, 99)( 37,100)( 38,101)( 39, 87)( 40, 88)( 41, 89)
( 42, 84)( 43, 85)( 44, 86)( 45, 90)( 46, 91)( 47, 92)( 48,105)( 49,106)
( 50,107)( 51,102)( 52,103)( 53,104)( 54,108)( 55,109)( 56,110);
s3 := 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);
s4 := Sym(110)!(  3,  4)(  6,  7)(  9, 10)( 12, 13)( 15, 16)( 18, 19)( 21, 22)
( 24, 25)( 27, 28)( 30, 31)( 33, 34)( 36, 37)( 39, 40)( 42, 43)( 45, 46)
( 48, 49)( 51, 52)( 54, 55)( 57, 58)( 60, 61)( 63, 64)( 66, 67)( 69, 70)
( 72, 73)( 75, 76)( 78, 79)( 81, 82)( 84, 85)( 87, 88)( 90, 91)( 93, 94)
( 96, 97)( 99,100)(102,103)(105,106)(108,109);
poly := sub<Sym(110)|s0,s1,s2,s3,s4>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

```

to this polytope