# Polytope of Type {39,6}

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

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

```
References : None.
to this polytope