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# Polytope of Type {6,36}

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