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# Polytope of Type {12,6,6}

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

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

```
References : None.
to this polytope