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# Polytope of Type {4,18,6}

Atlas Canonical Name : {4,18,6}*864b
if this polytope has a name.
Group : SmallGroup(864,2462)
Rank : 4
Schlafli Type : {4,18,6}
Number of vertices, edges, etc : 4, 36, 54, 6
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,18,6,2} of size 1728
Vertex Figure Of :
{2,4,18,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,18,6}*432b
3-fold quotients : {4,18,2}*288a, {4,6,6}*288b
4-fold quotients : {2,9,6}*216
6-fold quotients : {2,18,2}*144, {2,6,6}*144c
9-fold quotients : {4,6,2}*96a
12-fold quotients : {2,9,2}*72, {2,3,6}*72
18-fold quotients : {2,6,2}*48
27-fold quotients : {4,2,2}*32
36-fold quotients : {2,3,2}*24
54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,36,6}*1728b, {8,18,6}*1728b, {4,18,12}*1728b
Permutation Representation (GAP) :
```s0 := ( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)( 62, 89)
( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)( 70, 97)
( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)( 78,105)
( 79,106)( 80,107)( 81,108);;
s1 := (  1, 55)(  2, 57)(  3, 56)(  4, 61)(  5, 63)(  6, 62)(  7, 58)(  8, 60)
(  9, 59)( 10, 75)( 11, 74)( 12, 73)( 13, 81)( 14, 80)( 15, 79)( 16, 78)
( 17, 77)( 18, 76)( 19, 66)( 20, 65)( 21, 64)( 22, 72)( 23, 71)( 24, 70)
( 25, 69)( 26, 68)( 27, 67)( 28, 82)( 29, 84)( 30, 83)( 31, 88)( 32, 90)
( 33, 89)( 34, 85)( 35, 87)( 36, 86)( 37,102)( 38,101)( 39,100)( 40,108)
( 41,107)( 42,106)( 43,105)( 44,104)( 45,103)( 46, 93)( 47, 92)( 48, 91)
( 49, 99)( 50, 98)( 51, 97)( 52, 96)( 53, 95)( 54, 94);;
s2 := (  1, 13)(  2, 15)(  3, 14)(  4, 10)(  5, 12)(  6, 11)(  7, 16)(  8, 18)
(  9, 17)( 19, 24)( 20, 23)( 21, 22)( 25, 27)( 28, 40)( 29, 42)( 30, 41)
( 31, 37)( 32, 39)( 33, 38)( 34, 43)( 35, 45)( 36, 44)( 46, 51)( 47, 50)
( 48, 49)( 52, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)
( 61, 70)( 62, 72)( 63, 71)( 73, 78)( 74, 77)( 75, 76)( 79, 81)( 82, 94)
( 83, 96)( 84, 95)( 85, 91)( 86, 93)( 87, 92)( 88, 97)( 89, 99)( 90, 98)
(100,105)(101,104)(102,103)(106,108);;
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, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope