# Polytope of Type {6,39}

Atlas Canonical Name : {6,39}*468
if this polytope has a name.
Group : SmallGroup(468,45)
Rank : 3
Schlafli Type : {6,39}
Number of vertices, edges, etc : 6, 117, 39
Order of s0s1s2 : 78
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,39,2} of size 936
{6,39,4} of size 1872
Vertex Figure Of :
{2,6,39} of size 936
{3,6,39} of size 1404
{4,6,39} of size 1872
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,39}*156
9-fold quotients : {2,13}*52
13-fold quotients : {6,3}*36
39-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,78}*936c
3-fold covers : {6,117}*1404, {6,39}*1404
4-fold covers : {6,156}*1872c, {12,78}*1872c, {12,39}*1872, {6,39}*1872
Permutation Representation (GAP) :
```s0 := ( 40, 79)( 41, 80)( 42, 81)( 43, 82)( 44, 83)( 45, 84)( 46, 85)( 47, 86)
( 48, 87)( 49, 88)( 50, 89)( 51, 90)( 52, 91)( 53, 92)( 54, 93)( 55, 94)
( 56, 95)( 57, 96)( 58, 97)( 59, 98)( 60, 99)( 61,100)( 62,101)( 63,102)
( 64,103)( 65,104)( 66,105)( 67,106)( 68,107)( 69,108)( 70,109)( 71,110)
( 72,111)( 73,112)( 74,113)( 75,114)( 76,115)( 77,116)( 78,117);;
s1 := (  1, 40)(  2, 52)(  3, 51)(  4, 50)(  5, 49)(  6, 48)(  7, 47)(  8, 46)
(  9, 45)( 10, 44)( 11, 43)( 12, 42)( 13, 41)( 14, 66)( 15, 78)( 16, 77)
( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 71)( 23, 70)( 24, 69)
( 25, 68)( 26, 67)( 27, 53)( 28, 65)( 29, 64)( 30, 63)( 31, 62)( 32, 61)
( 33, 60)( 34, 59)( 35, 58)( 36, 57)( 37, 56)( 38, 55)( 39, 54)( 80, 91)
( 81, 90)( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 92,105)( 93,117)( 94,116)
( 95,115)( 96,114)( 97,113)( 98,112)( 99,111)(100,110)(101,109)(102,108)
(103,107)(104,106);;
s2 := (  1, 15)(  2, 14)(  3, 26)(  4, 25)(  5, 24)(  6, 23)(  7, 22)(  8, 21)
(  9, 20)( 10, 19)( 11, 18)( 12, 17)( 13, 16)( 27, 28)( 29, 39)( 30, 38)
( 31, 37)( 32, 36)( 33, 35)( 40, 93)( 41, 92)( 42,104)( 43,103)( 44,102)
( 45,101)( 46,100)( 47, 99)( 48, 98)( 49, 97)( 50, 96)( 51, 95)( 52, 94)
( 53, 80)( 54, 79)( 55, 91)( 56, 90)( 57, 89)( 58, 88)( 59, 87)( 60, 86)
( 61, 85)( 62, 84)( 63, 83)( 64, 82)( 65, 81)( 66,106)( 67,105)( 68,117)
( 69,116)( 70,115)( 71,114)( 72,113)( 73,112)( 74,111)( 75,110)( 76,109)
( 77,108)( 78,107);;
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*s0*s1*s2*s0*s1*s0*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*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(117)!( 40, 79)( 41, 80)( 42, 81)( 43, 82)( 44, 83)( 45, 84)( 46, 85)
( 47, 86)( 48, 87)( 49, 88)( 50, 89)( 51, 90)( 52, 91)( 53, 92)( 54, 93)
( 55, 94)( 56, 95)( 57, 96)( 58, 97)( 59, 98)( 60, 99)( 61,100)( 62,101)
( 63,102)( 64,103)( 65,104)( 66,105)( 67,106)( 68,107)( 69,108)( 70,109)
( 71,110)( 72,111)( 73,112)( 74,113)( 75,114)( 76,115)( 77,116)( 78,117);
s1 := Sym(117)!(  1, 40)(  2, 52)(  3, 51)(  4, 50)(  5, 49)(  6, 48)(  7, 47)
(  8, 46)(  9, 45)( 10, 44)( 11, 43)( 12, 42)( 13, 41)( 14, 66)( 15, 78)
( 16, 77)( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 71)( 23, 70)
( 24, 69)( 25, 68)( 26, 67)( 27, 53)( 28, 65)( 29, 64)( 30, 63)( 31, 62)
( 32, 61)( 33, 60)( 34, 59)( 35, 58)( 36, 57)( 37, 56)( 38, 55)( 39, 54)
( 80, 91)( 81, 90)( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 92,105)( 93,117)
( 94,116)( 95,115)( 96,114)( 97,113)( 98,112)( 99,111)(100,110)(101,109)
(102,108)(103,107)(104,106);
s2 := Sym(117)!(  1, 15)(  2, 14)(  3, 26)(  4, 25)(  5, 24)(  6, 23)(  7, 22)
(  8, 21)(  9, 20)( 10, 19)( 11, 18)( 12, 17)( 13, 16)( 27, 28)( 29, 39)
( 30, 38)( 31, 37)( 32, 36)( 33, 35)( 40, 93)( 41, 92)( 42,104)( 43,103)
( 44,102)( 45,101)( 46,100)( 47, 99)( 48, 98)( 49, 97)( 50, 96)( 51, 95)
( 52, 94)( 53, 80)( 54, 79)( 55, 91)( 56, 90)( 57, 89)( 58, 88)( 59, 87)
( 60, 86)( 61, 85)( 62, 84)( 63, 83)( 64, 82)( 65, 81)( 66,106)( 67,105)
( 68,117)( 69,116)( 70,115)( 71,114)( 72,113)( 73,112)( 74,111)( 75,110)
( 76,109)( 77,108)( 78,107);
poly := sub<Sym(117)|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*s0*s1*s2*s0*s1*s0*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*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope