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# Polytope of Type {4,60}

Atlas Canonical Name : {4,60}*480c
if this polytope has a name.
Group : SmallGroup(480,1026)
Rank : 3
Schlafli Type : {4,60}
Number of vertices, edges, etc : 4, 120, 60
Order of s0s1s2 : 60
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,60,2} of size 960
{4,60,4} of size 1920
{4,60,4} of size 1920
{4,60,4} of size 1920
Vertex Figure Of :
{2,4,60} of size 960
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,30}*240b
4-fold quotients : {4,15}*120
5-fold quotients : {4,12}*96c
10-fold quotients : {4,6}*48c
20-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,60}*960b
3-fold covers : {4,180}*1440c
4-fold covers : {4,60}*1920c, {4,60}*1920d, {8,60}*1920e, {8,60}*1920f, {4,120}*1920c, {4,120}*1920d
Permutation Representation (GAP) :
```s0 := (  1, 63)(  2, 64)(  3, 61)(  4, 62)(  5, 67)(  6, 68)(  7, 65)(  8, 66)
(  9, 71)( 10, 72)( 11, 69)( 12, 70)( 13, 75)( 14, 76)( 15, 73)( 16, 74)
( 17, 79)( 18, 80)( 19, 77)( 20, 78)( 21, 83)( 22, 84)( 23, 81)( 24, 82)
( 25, 87)( 26, 88)( 27, 85)( 28, 86)( 29, 91)( 30, 92)( 31, 89)( 32, 90)
( 33, 95)( 34, 96)( 35, 93)( 36, 94)( 37, 99)( 38,100)( 39, 97)( 40, 98)
( 41,103)( 42,104)( 43,101)( 44,102)( 45,107)( 46,108)( 47,105)( 48,106)
( 49,111)( 50,112)( 51,109)( 52,110)( 53,115)( 54,116)( 55,113)( 56,114)
( 57,119)( 58,120)( 59,117)( 60,118)(121,183)(122,184)(123,181)(124,182)
(125,187)(126,188)(127,185)(128,186)(129,191)(130,192)(131,189)(132,190)
(133,195)(134,196)(135,193)(136,194)(137,199)(138,200)(139,197)(140,198)
(141,203)(142,204)(143,201)(144,202)(145,207)(146,208)(147,205)(148,206)
(149,211)(150,212)(151,209)(152,210)(153,215)(154,216)(155,213)(156,214)
(157,219)(158,220)(159,217)(160,218)(161,223)(162,224)(163,221)(164,222)
(165,227)(166,228)(167,225)(168,226)(169,231)(170,232)(171,229)(172,230)
(173,235)(174,236)(175,233)(176,234)(177,239)(178,240)(179,237)(180,238);;
s1 := (  2,  3)(  5, 17)(  6, 19)(  7, 18)(  8, 20)(  9, 13)( 10, 15)( 11, 14)
( 12, 16)( 21, 41)( 22, 43)( 23, 42)( 24, 44)( 25, 57)( 26, 59)( 27, 58)
( 28, 60)( 29, 53)( 30, 55)( 31, 54)( 32, 56)( 33, 49)( 34, 51)( 35, 50)
( 36, 52)( 37, 45)( 38, 47)( 39, 46)( 40, 48)( 62, 63)( 65, 77)( 66, 79)
( 67, 78)( 68, 80)( 69, 73)( 70, 75)( 71, 74)( 72, 76)( 81,101)( 82,103)
( 83,102)( 84,104)( 85,117)( 86,119)( 87,118)( 88,120)( 89,113)( 90,115)
( 91,114)( 92,116)( 93,109)( 94,111)( 95,110)( 96,112)( 97,105)( 98,107)
( 99,106)(100,108)(121,181)(122,183)(123,182)(124,184)(125,197)(126,199)
(127,198)(128,200)(129,193)(130,195)(131,194)(132,196)(133,189)(134,191)
(135,190)(136,192)(137,185)(138,187)(139,186)(140,188)(141,221)(142,223)
(143,222)(144,224)(145,237)(146,239)(147,238)(148,240)(149,233)(150,235)
(151,234)(152,236)(153,229)(154,231)(155,230)(156,232)(157,225)(158,227)
(159,226)(160,228)(161,201)(162,203)(163,202)(164,204)(165,217)(166,219)
(167,218)(168,220)(169,213)(170,215)(171,214)(172,216)(173,209)(174,211)
(175,210)(176,212)(177,205)(178,207)(179,206)(180,208);;
s2 := (  1,165)(  2,168)(  3,167)(  4,166)(  5,161)(  6,164)(  7,163)(  8,162)
(  9,177)( 10,180)( 11,179)( 12,178)( 13,173)( 14,176)( 15,175)( 16,174)
( 17,169)( 18,172)( 19,171)( 20,170)( 21,145)( 22,148)( 23,147)( 24,146)
( 25,141)( 26,144)( 27,143)( 28,142)( 29,157)( 30,160)( 31,159)( 32,158)
( 33,153)( 34,156)( 35,155)( 36,154)( 37,149)( 38,152)( 39,151)( 40,150)
( 41,125)( 42,128)( 43,127)( 44,126)( 45,121)( 46,124)( 47,123)( 48,122)
( 49,137)( 50,140)( 51,139)( 52,138)( 53,133)( 54,136)( 55,135)( 56,134)
( 57,129)( 58,132)( 59,131)( 60,130)( 61,225)( 62,228)( 63,227)( 64,226)
( 65,221)( 66,224)( 67,223)( 68,222)( 69,237)( 70,240)( 71,239)( 72,238)
( 73,233)( 74,236)( 75,235)( 76,234)( 77,229)( 78,232)( 79,231)( 80,230)
( 81,205)( 82,208)( 83,207)( 84,206)( 85,201)( 86,204)( 87,203)( 88,202)
( 89,217)( 90,220)( 91,219)( 92,218)( 93,213)( 94,216)( 95,215)( 96,214)
( 97,209)( 98,212)( 99,211)(100,210)(101,185)(102,188)(103,187)(104,186)
(105,181)(106,184)(107,183)(108,182)(109,197)(110,200)(111,199)(112,198)
(113,193)(114,196)(115,195)(116,194)(117,189)(118,192)(119,191)(120,190);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*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*s0*s2*s1*s0*s2*s1*s0*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(240)!(  1, 63)(  2, 64)(  3, 61)(  4, 62)(  5, 67)(  6, 68)(  7, 65)
(  8, 66)(  9, 71)( 10, 72)( 11, 69)( 12, 70)( 13, 75)( 14, 76)( 15, 73)
( 16, 74)( 17, 79)( 18, 80)( 19, 77)( 20, 78)( 21, 83)( 22, 84)( 23, 81)
( 24, 82)( 25, 87)( 26, 88)( 27, 85)( 28, 86)( 29, 91)( 30, 92)( 31, 89)
( 32, 90)( 33, 95)( 34, 96)( 35, 93)( 36, 94)( 37, 99)( 38,100)( 39, 97)
( 40, 98)( 41,103)( 42,104)( 43,101)( 44,102)( 45,107)( 46,108)( 47,105)
( 48,106)( 49,111)( 50,112)( 51,109)( 52,110)( 53,115)( 54,116)( 55,113)
( 56,114)( 57,119)( 58,120)( 59,117)( 60,118)(121,183)(122,184)(123,181)
(124,182)(125,187)(126,188)(127,185)(128,186)(129,191)(130,192)(131,189)
(132,190)(133,195)(134,196)(135,193)(136,194)(137,199)(138,200)(139,197)
(140,198)(141,203)(142,204)(143,201)(144,202)(145,207)(146,208)(147,205)
(148,206)(149,211)(150,212)(151,209)(152,210)(153,215)(154,216)(155,213)
(156,214)(157,219)(158,220)(159,217)(160,218)(161,223)(162,224)(163,221)
(164,222)(165,227)(166,228)(167,225)(168,226)(169,231)(170,232)(171,229)
(172,230)(173,235)(174,236)(175,233)(176,234)(177,239)(178,240)(179,237)
(180,238);
s1 := Sym(240)!(  2,  3)(  5, 17)(  6, 19)(  7, 18)(  8, 20)(  9, 13)( 10, 15)
( 11, 14)( 12, 16)( 21, 41)( 22, 43)( 23, 42)( 24, 44)( 25, 57)( 26, 59)
( 27, 58)( 28, 60)( 29, 53)( 30, 55)( 31, 54)( 32, 56)( 33, 49)( 34, 51)
( 35, 50)( 36, 52)( 37, 45)( 38, 47)( 39, 46)( 40, 48)( 62, 63)( 65, 77)
( 66, 79)( 67, 78)( 68, 80)( 69, 73)( 70, 75)( 71, 74)( 72, 76)( 81,101)
( 82,103)( 83,102)( 84,104)( 85,117)( 86,119)( 87,118)( 88,120)( 89,113)
( 90,115)( 91,114)( 92,116)( 93,109)( 94,111)( 95,110)( 96,112)( 97,105)
( 98,107)( 99,106)(100,108)(121,181)(122,183)(123,182)(124,184)(125,197)
(126,199)(127,198)(128,200)(129,193)(130,195)(131,194)(132,196)(133,189)
(134,191)(135,190)(136,192)(137,185)(138,187)(139,186)(140,188)(141,221)
(142,223)(143,222)(144,224)(145,237)(146,239)(147,238)(148,240)(149,233)
(150,235)(151,234)(152,236)(153,229)(154,231)(155,230)(156,232)(157,225)
(158,227)(159,226)(160,228)(161,201)(162,203)(163,202)(164,204)(165,217)
(166,219)(167,218)(168,220)(169,213)(170,215)(171,214)(172,216)(173,209)
(174,211)(175,210)(176,212)(177,205)(178,207)(179,206)(180,208);
s2 := Sym(240)!(  1,165)(  2,168)(  3,167)(  4,166)(  5,161)(  6,164)(  7,163)
(  8,162)(  9,177)( 10,180)( 11,179)( 12,178)( 13,173)( 14,176)( 15,175)
( 16,174)( 17,169)( 18,172)( 19,171)( 20,170)( 21,145)( 22,148)( 23,147)
( 24,146)( 25,141)( 26,144)( 27,143)( 28,142)( 29,157)( 30,160)( 31,159)
( 32,158)( 33,153)( 34,156)( 35,155)( 36,154)( 37,149)( 38,152)( 39,151)
( 40,150)( 41,125)( 42,128)( 43,127)( 44,126)( 45,121)( 46,124)( 47,123)
( 48,122)( 49,137)( 50,140)( 51,139)( 52,138)( 53,133)( 54,136)( 55,135)
( 56,134)( 57,129)( 58,132)( 59,131)( 60,130)( 61,225)( 62,228)( 63,227)
( 64,226)( 65,221)( 66,224)( 67,223)( 68,222)( 69,237)( 70,240)( 71,239)
( 72,238)( 73,233)( 74,236)( 75,235)( 76,234)( 77,229)( 78,232)( 79,231)
( 80,230)( 81,205)( 82,208)( 83,207)( 84,206)( 85,201)( 86,204)( 87,203)
( 88,202)( 89,217)( 90,220)( 91,219)( 92,218)( 93,213)( 94,216)( 95,215)
( 96,214)( 97,209)( 98,212)( 99,211)(100,210)(101,185)(102,188)(103,187)
(104,186)(105,181)(106,184)(107,183)(108,182)(109,197)(110,200)(111,199)
(112,198)(113,193)(114,196)(115,195)(116,194)(117,189)(118,192)(119,191)
(120,190);
poly := sub<Sym(240)|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*s2*s1*s2*s1*s0*s1*s2*s1*s2*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*s0*s2*s1*s0*s2*s1*s0*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