# Polytope of Type {6,60}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,60}*720b
Also Known As : {6,60|2}. if this polytope has another name.
Group : SmallGroup(720,673)
Rank : 3
Schlafli Type : {6,60}
Number of vertices, edges, etc : 6, 180, 60
Order of s0s1s2 : 60
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,60,2} of size 1440
Vertex Figure Of :
{2,6,60} of size 1440
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,30}*360b
3-fold quotients : {6,20}*240a, {2,60}*240
5-fold quotients : {6,12}*144a
6-fold quotients : {6,10}*120, {2,30}*120
9-fold quotients : {2,20}*80
10-fold quotients : {6,6}*72a
12-fold quotients : {2,15}*60
15-fold quotients : {2,12}*48, {6,4}*48a
18-fold quotients : {2,10}*40
30-fold quotients : {2,6}*24, {6,2}*24
36-fold quotients : {2,5}*20
45-fold quotients : {2,4}*16
60-fold quotients : {2,3}*12, {3,2}*12
90-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,120}*1440b, {12,60}*1440b
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)( 61, 76)
( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)( 69, 84)
( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)(106,121)(107,122)
(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)(115,130)
(116,131)(117,132)(118,133)(119,134)(120,135)(151,166)(152,167)(153,168)
(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)
(162,177)(163,178)(164,179)(165,180);;
s1 := (  1, 16)(  2, 20)(  3, 19)(  4, 18)(  5, 17)(  6, 26)(  7, 30)(  8, 29)
(  9, 28)( 10, 27)( 11, 21)( 12, 25)( 13, 24)( 14, 23)( 15, 22)( 32, 35)
( 33, 34)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 61)( 47, 65)
( 48, 64)( 49, 63)( 50, 62)( 51, 71)( 52, 75)( 53, 74)( 54, 73)( 55, 72)
( 56, 66)( 57, 70)( 58, 69)( 59, 68)( 60, 67)( 77, 80)( 78, 79)( 81, 86)
( 82, 90)( 83, 89)( 84, 88)( 85, 87)( 91,151)( 92,155)( 93,154)( 94,153)
( 95,152)( 96,161)( 97,165)( 98,164)( 99,163)(100,162)(101,156)(102,160)
(103,159)(104,158)(105,157)(106,136)(107,140)(108,139)(109,138)(110,137)
(111,146)(112,150)(113,149)(114,148)(115,147)(116,141)(117,145)(118,144)
(119,143)(120,142)(121,166)(122,170)(123,169)(124,168)(125,167)(126,176)
(127,180)(128,179)(129,178)(130,177)(131,171)(132,175)(133,174)(134,173)
(135,172);;
s2 := (  1, 97)(  2, 96)(  3,100)(  4, 99)(  5, 98)(  6, 92)(  7, 91)(  8, 95)
(  9, 94)( 10, 93)( 11,102)( 12,101)( 13,105)( 14,104)( 15,103)( 16,112)
( 17,111)( 18,115)( 19,114)( 20,113)( 21,107)( 22,106)( 23,110)( 24,109)
( 25,108)( 26,117)( 27,116)( 28,120)( 29,119)( 30,118)( 31,127)( 32,126)
( 33,130)( 34,129)( 35,128)( 36,122)( 37,121)( 38,125)( 39,124)( 40,123)
( 41,132)( 42,131)( 43,135)( 44,134)( 45,133)( 46,142)( 47,141)( 48,145)
( 49,144)( 50,143)( 51,137)( 52,136)( 53,140)( 54,139)( 55,138)( 56,147)
( 57,146)( 58,150)( 59,149)( 60,148)( 61,157)( 62,156)( 63,160)( 64,159)
( 65,158)( 66,152)( 67,151)( 68,155)( 69,154)( 70,153)( 71,162)( 72,161)
( 73,165)( 74,164)( 75,163)( 76,172)( 77,171)( 78,175)( 79,174)( 80,173)
( 81,167)( 82,166)( 83,170)( 84,169)( 85,168)( 86,177)( 87,176)( 88,180)
( 89,179)( 90,178);;
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*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(180)!( 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)
( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)
( 69, 84)( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)(106,121)
(107,122)(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)
(115,130)(116,131)(117,132)(118,133)(119,134)(120,135)(151,166)(152,167)
(153,168)(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)
(161,176)(162,177)(163,178)(164,179)(165,180);
s1 := Sym(180)!(  1, 16)(  2, 20)(  3, 19)(  4, 18)(  5, 17)(  6, 26)(  7, 30)
(  8, 29)(  9, 28)( 10, 27)( 11, 21)( 12, 25)( 13, 24)( 14, 23)( 15, 22)
( 32, 35)( 33, 34)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 61)
( 47, 65)( 48, 64)( 49, 63)( 50, 62)( 51, 71)( 52, 75)( 53, 74)( 54, 73)
( 55, 72)( 56, 66)( 57, 70)( 58, 69)( 59, 68)( 60, 67)( 77, 80)( 78, 79)
( 81, 86)( 82, 90)( 83, 89)( 84, 88)( 85, 87)( 91,151)( 92,155)( 93,154)
( 94,153)( 95,152)( 96,161)( 97,165)( 98,164)( 99,163)(100,162)(101,156)
(102,160)(103,159)(104,158)(105,157)(106,136)(107,140)(108,139)(109,138)
(110,137)(111,146)(112,150)(113,149)(114,148)(115,147)(116,141)(117,145)
(118,144)(119,143)(120,142)(121,166)(122,170)(123,169)(124,168)(125,167)
(126,176)(127,180)(128,179)(129,178)(130,177)(131,171)(132,175)(133,174)
(134,173)(135,172);
s2 := Sym(180)!(  1, 97)(  2, 96)(  3,100)(  4, 99)(  5, 98)(  6, 92)(  7, 91)
(  8, 95)(  9, 94)( 10, 93)( 11,102)( 12,101)( 13,105)( 14,104)( 15,103)
( 16,112)( 17,111)( 18,115)( 19,114)( 20,113)( 21,107)( 22,106)( 23,110)
( 24,109)( 25,108)( 26,117)( 27,116)( 28,120)( 29,119)( 30,118)( 31,127)
( 32,126)( 33,130)( 34,129)( 35,128)( 36,122)( 37,121)( 38,125)( 39,124)
( 40,123)( 41,132)( 42,131)( 43,135)( 44,134)( 45,133)( 46,142)( 47,141)
( 48,145)( 49,144)( 50,143)( 51,137)( 52,136)( 53,140)( 54,139)( 55,138)
( 56,147)( 57,146)( 58,150)( 59,149)( 60,148)( 61,157)( 62,156)( 63,160)
( 64,159)( 65,158)( 66,152)( 67,151)( 68,155)( 69,154)( 70,153)( 71,162)
( 72,161)( 73,165)( 74,164)( 75,163)( 76,172)( 77,171)( 78,175)( 79,174)
( 80,173)( 81,167)( 82,166)( 83,170)( 84,169)( 85,168)( 86,177)( 87,176)
( 88,180)( 89,179)( 90,178);
poly := sub<Sym(180)|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*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