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# Polytope of Type {36,6}

Atlas Canonical Name : {36,6}*432b
if this polytope has a name.
Group : SmallGroup(432,291)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 36, 108, 6
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{36,6,2} of size 864
{36,6,3} of size 1296
{36,6,4} of size 1728
Vertex Figure Of :
{2,36,6} of size 864
{4,36,6} of size 1728
{4,36,6} of size 1728
{4,36,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,6}*216b
3-fold quotients : {36,2}*144, {12,6}*144b
4-fold quotients : {9,6}*108
6-fold quotients : {18,2}*72, {6,6}*72c
9-fold quotients : {12,2}*48
12-fold quotients : {9,2}*36, {3,6}*36
18-fold quotients : {6,2}*24
27-fold quotients : {4,2}*16
36-fold quotients : {3,2}*12
54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {72,6}*864b, {36,12}*864b
3-fold covers : {36,18}*1296b, {36,6}*1296a, {108,6}*1296b, {36,6}*1296l
4-fold covers : {144,6}*1728b, {36,24}*1728a, {36,12}*1728b, {36,24}*1728b, {72,12}*1728b, {72,12}*1728d, {36,6}*1728a, {36,12}*1728f
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 20)( 11, 19)( 12, 21)( 13, 26)
( 14, 25)( 15, 27)( 16, 23)( 17, 22)( 18, 24)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 47)( 38, 46)( 39, 48)( 40, 53)( 41, 52)( 42, 54)( 43, 50)
( 44, 49)( 45, 51)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)
( 61, 85)( 62, 87)( 63, 86)( 64,101)( 65,100)( 66,102)( 67,107)( 68,106)
( 69,108)( 70,104)( 71,103)( 72,105)( 73, 92)( 74, 91)( 75, 93)( 76, 98)
( 77, 97)( 78, 99)( 79, 95)( 80, 94)( 81, 96);;
s1 := (  1, 67)(  2, 69)(  3, 68)(  4, 64)(  5, 66)(  6, 65)(  7, 70)(  8, 72)
(  9, 71)( 10, 58)( 11, 60)( 12, 59)( 13, 55)( 14, 57)( 15, 56)( 16, 61)
( 17, 63)( 18, 62)( 19, 77)( 20, 76)( 21, 78)( 22, 74)( 23, 73)( 24, 75)
( 25, 80)( 26, 79)( 27, 81)( 28, 94)( 29, 96)( 30, 95)( 31, 91)( 32, 93)
( 33, 92)( 34, 97)( 35, 99)( 36, 98)( 37, 85)( 38, 87)( 39, 86)( 40, 82)
( 41, 84)( 42, 83)( 43, 88)( 44, 90)( 45, 89)( 46,104)( 47,103)( 48,105)
( 49,101)( 50,100)( 51,102)( 52,107)( 53,106)( 54,108);;
s2 := (  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]);;

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

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