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# Polytope of Type {6,60}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,60}*1080c
if this polytope has a name.
Group : SmallGroup(1080,528)
Rank : 3
Schlafli Type : {6,60}
Number of vertices, edges, etc : 9, 270, 90
Order of s0s1s2 : 60
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,20}*360
5-fold quotients : {6,12}*216c
15-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := ( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)( 23, 38)
( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 46, 91)
( 47, 92)( 48, 93)( 49, 94)( 50, 95)( 51, 96)( 52, 97)( 53, 98)( 54, 99)
( 55,100)( 56,101)( 57,102)( 58,103)( 59,104)( 60,105)( 61,121)( 62,122)
( 63,123)( 64,124)( 65,125)( 66,126)( 67,127)( 68,128)( 69,129)( 70,130)
( 71,131)( 72,132)( 73,133)( 74,134)( 75,135)( 76,106)( 77,107)( 78,108)
( 79,109)( 80,110)( 81,111)( 82,112)( 83,113)( 84,114)( 85,115)( 86,116)
( 87,117)( 88,118)( 89,119)( 90,120);;
s1 := (  1, 46)(  2, 50)(  3, 49)(  4, 48)(  5, 47)(  6, 56)(  7, 60)(  8, 59)
(  9, 58)( 10, 57)( 11, 51)( 12, 55)( 13, 54)( 14, 53)( 15, 52)( 16, 61)
( 17, 65)( 18, 64)( 19, 63)( 20, 62)( 21, 71)( 22, 75)( 23, 74)( 24, 73)
( 25, 72)( 26, 66)( 27, 70)( 28, 69)( 29, 68)( 30, 67)( 31, 76)( 32, 80)
( 33, 79)( 34, 78)( 35, 77)( 36, 86)( 37, 90)( 38, 89)( 39, 88)( 40, 87)
( 41, 81)( 42, 85)( 43, 84)( 44, 83)( 45, 82)( 92, 95)( 93, 94)( 96,101)
( 97,105)( 98,104)( 99,103)(100,102)(107,110)(108,109)(111,116)(112,120)
(113,119)(114,118)(115,117)(122,125)(123,124)(126,131)(127,135)(128,134)
(129,133)(130,132);;
s2 := (  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)( 16, 52)
( 17, 51)( 18, 55)( 19, 54)( 20, 53)( 21, 47)( 22, 46)( 23, 50)( 24, 49)
( 25, 48)( 26, 57)( 27, 56)( 28, 60)( 29, 59)( 30, 58)( 31, 97)( 32, 96)
( 33,100)( 34, 99)( 35, 98)( 36, 92)( 37, 91)( 38, 95)( 39, 94)( 40, 93)
( 41,102)( 42,101)( 43,105)( 44,104)( 45,103)( 61, 67)( 62, 66)( 63, 70)
( 64, 69)( 65, 68)( 71, 72)( 73, 75)( 76,112)( 77,111)( 78,115)( 79,114)
( 80,113)( 81,107)( 82,106)( 83,110)( 84,109)( 85,108)( 86,117)( 87,116)
( 88,120)( 89,119)( 90,118)(121,127)(122,126)(123,130)(124,129)(125,128)
(131,132)(133,135);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*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(135)!( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)
( 23, 38)( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)
( 46, 91)( 47, 92)( 48, 93)( 49, 94)( 50, 95)( 51, 96)( 52, 97)( 53, 98)
( 54, 99)( 55,100)( 56,101)( 57,102)( 58,103)( 59,104)( 60,105)( 61,121)
( 62,122)( 63,123)( 64,124)( 65,125)( 66,126)( 67,127)( 68,128)( 69,129)
( 70,130)( 71,131)( 72,132)( 73,133)( 74,134)( 75,135)( 76,106)( 77,107)
( 78,108)( 79,109)( 80,110)( 81,111)( 82,112)( 83,113)( 84,114)( 85,115)
( 86,116)( 87,117)( 88,118)( 89,119)( 90,120);
s1 := Sym(135)!(  1, 46)(  2, 50)(  3, 49)(  4, 48)(  5, 47)(  6, 56)(  7, 60)
(  8, 59)(  9, 58)( 10, 57)( 11, 51)( 12, 55)( 13, 54)( 14, 53)( 15, 52)
( 16, 61)( 17, 65)( 18, 64)( 19, 63)( 20, 62)( 21, 71)( 22, 75)( 23, 74)
( 24, 73)( 25, 72)( 26, 66)( 27, 70)( 28, 69)( 29, 68)( 30, 67)( 31, 76)
( 32, 80)( 33, 79)( 34, 78)( 35, 77)( 36, 86)( 37, 90)( 38, 89)( 39, 88)
( 40, 87)( 41, 81)( 42, 85)( 43, 84)( 44, 83)( 45, 82)( 92, 95)( 93, 94)
( 96,101)( 97,105)( 98,104)( 99,103)(100,102)(107,110)(108,109)(111,116)
(112,120)(113,119)(114,118)(115,117)(122,125)(123,124)(126,131)(127,135)
(128,134)(129,133)(130,132);
s2 := Sym(135)!(  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)
( 16, 52)( 17, 51)( 18, 55)( 19, 54)( 20, 53)( 21, 47)( 22, 46)( 23, 50)
( 24, 49)( 25, 48)( 26, 57)( 27, 56)( 28, 60)( 29, 59)( 30, 58)( 31, 97)
( 32, 96)( 33,100)( 34, 99)( 35, 98)( 36, 92)( 37, 91)( 38, 95)( 39, 94)
( 40, 93)( 41,102)( 42,101)( 43,105)( 44,104)( 45,103)( 61, 67)( 62, 66)
( 63, 70)( 64, 69)( 65, 68)( 71, 72)( 73, 75)( 76,112)( 77,111)( 78,115)
( 79,114)( 80,113)( 81,107)( 82,106)( 83,110)( 84,109)( 85,108)( 86,117)
( 87,116)( 88,120)( 89,119)( 90,118)(121,127)(122,126)(123,130)(124,129)
(125,128)(131,132)(133,135);
poly := sub<Sym(135)|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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*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