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# Polytope of Type {60,6}

Atlas Canonical Name : {60,6}*1440d
if this polytope has a name.
Group : SmallGroup(1440,5901)
Rank : 3
Schlafli Type : {60,6}
Number of vertices, edges, etc : 120, 360, 12
Order of s0s1s2 : 30
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {60,6}*720d
3-fold quotients : {20,6}*480c
4-fold quotients : {30,6}*360b
5-fold quotients : {12,6}*288a
6-fold quotients : {20,6}*240b
10-fold quotients : {12,6}*144d
12-fold quotients : {10,6}*120, {30,2}*120
15-fold quotients : {4,6}*96
20-fold quotients : {6,6}*72a
24-fold quotients : {15,2}*60
30-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
36-fold quotients : {10,2}*40
60-fold quotients : {4,3}*24, {2,6}*24, {6,2}*24
72-fold quotients : {5,2}*20
120-fold quotients : {2,3}*12, {3,2}*12
180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5, 19)(  6, 20)(  7, 17)(  8, 18)(  9, 15)( 10, 16)
( 11, 13)( 12, 14)( 21, 43)( 22, 44)( 23, 41)( 24, 42)( 25, 59)( 26, 60)
( 27, 57)( 28, 58)( 29, 55)( 30, 56)( 31, 53)( 32, 54)( 33, 51)( 34, 52)
( 35, 49)( 36, 50)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 61, 63)( 62, 64)
( 65, 79)( 66, 80)( 67, 77)( 68, 78)( 69, 75)( 70, 76)( 71, 73)( 72, 74)
( 81,103)( 82,104)( 83,101)( 84,102)( 85,119)( 86,120)( 87,117)( 88,118)
( 89,115)( 90,116)( 91,113)( 92,114)( 93,111)( 94,112)( 95,109)( 96,110)
( 97,107)( 98,108)( 99,105)(100,106)(121,123)(122,124)(125,139)(126,140)
(127,137)(128,138)(129,135)(130,136)(131,133)(132,134)(141,163)(142,164)
(143,161)(144,162)(145,179)(146,180)(147,177)(148,178)(149,175)(150,176)
(151,173)(152,174)(153,171)(154,172)(155,169)(156,170)(157,167)(158,168)
(159,165)(160,166)(181,183)(182,184)(185,199)(186,200)(187,197)(188,198)
(189,195)(190,196)(191,193)(192,194)(201,223)(202,224)(203,221)(204,222)
(205,239)(206,240)(207,237)(208,238)(209,235)(210,236)(211,233)(212,234)
(213,231)(214,232)(215,229)(216,230)(217,227)(218,228)(219,225)(220,226)
(241,243)(242,244)(245,259)(246,260)(247,257)(248,258)(249,255)(250,256)
(251,253)(252,254)(261,283)(262,284)(263,281)(264,282)(265,299)(266,300)
(267,297)(268,298)(269,295)(270,296)(271,293)(272,294)(273,291)(274,292)
(275,289)(276,290)(277,287)(278,288)(279,285)(280,286)(301,303)(302,304)
(305,319)(306,320)(307,317)(308,318)(309,315)(310,316)(311,313)(312,314)
(321,343)(322,344)(323,341)(324,342)(325,359)(326,360)(327,357)(328,358)
(329,355)(330,356)(331,353)(332,354)(333,351)(334,352)(335,349)(336,350)
(337,347)(338,348)(339,345)(340,346);;
s1 := (  1, 25)(  2, 26)(  3, 28)(  4, 27)(  5, 21)(  6, 22)(  7, 24)(  8, 23)
(  9, 37)( 10, 38)( 11, 40)( 12, 39)( 13, 33)( 14, 34)( 15, 36)( 16, 35)
( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 41, 45)( 42, 46)( 43, 48)( 44, 47)
( 49, 57)( 50, 58)( 51, 60)( 52, 59)( 55, 56)( 61,145)( 62,146)( 63,148)
( 64,147)( 65,141)( 66,142)( 67,144)( 68,143)( 69,157)( 70,158)( 71,160)
( 72,159)( 73,153)( 74,154)( 75,156)( 76,155)( 77,149)( 78,150)( 79,152)
( 80,151)( 81,125)( 82,126)( 83,128)( 84,127)( 85,121)( 86,122)( 87,124)
( 88,123)( 89,137)( 90,138)( 91,140)( 92,139)( 93,133)( 94,134)( 95,136)
( 96,135)( 97,129)( 98,130)( 99,132)(100,131)(101,165)(102,166)(103,168)
(104,167)(105,161)(106,162)(107,164)(108,163)(109,177)(110,178)(111,180)
(112,179)(113,173)(114,174)(115,176)(116,175)(117,169)(118,170)(119,172)
(120,171)(181,205)(182,206)(183,208)(184,207)(185,201)(186,202)(187,204)
(188,203)(189,217)(190,218)(191,220)(192,219)(193,213)(194,214)(195,216)
(196,215)(197,209)(198,210)(199,212)(200,211)(221,225)(222,226)(223,228)
(224,227)(229,237)(230,238)(231,240)(232,239)(235,236)(241,325)(242,326)
(243,328)(244,327)(245,321)(246,322)(247,324)(248,323)(249,337)(250,338)
(251,340)(252,339)(253,333)(254,334)(255,336)(256,335)(257,329)(258,330)
(259,332)(260,331)(261,305)(262,306)(263,308)(264,307)(265,301)(266,302)
(267,304)(268,303)(269,317)(270,318)(271,320)(272,319)(273,313)(274,314)
(275,316)(276,315)(277,309)(278,310)(279,312)(280,311)(281,345)(282,346)
(283,348)(284,347)(285,341)(286,342)(287,344)(288,343)(289,357)(290,358)
(291,360)(292,359)(293,353)(294,354)(295,356)(296,355)(297,349)(298,350)
(299,352)(300,351);;
s2 := (  1,241)(  2,244)(  3,243)(  4,242)(  5,245)(  6,248)(  7,247)(  8,246)
(  9,249)( 10,252)( 11,251)( 12,250)( 13,253)( 14,256)( 15,255)( 16,254)
( 17,257)( 18,260)( 19,259)( 20,258)( 21,261)( 22,264)( 23,263)( 24,262)
( 25,265)( 26,268)( 27,267)( 28,266)( 29,269)( 30,272)( 31,271)( 32,270)
( 33,273)( 34,276)( 35,275)( 36,274)( 37,277)( 38,280)( 39,279)( 40,278)
( 41,281)( 42,284)( 43,283)( 44,282)( 45,285)( 46,288)( 47,287)( 48,286)
( 49,289)( 50,292)( 51,291)( 52,290)( 53,293)( 54,296)( 55,295)( 56,294)
( 57,297)( 58,300)( 59,299)( 60,298)( 61,181)( 62,184)( 63,183)( 64,182)
( 65,185)( 66,188)( 67,187)( 68,186)( 69,189)( 70,192)( 71,191)( 72,190)
( 73,193)( 74,196)( 75,195)( 76,194)( 77,197)( 78,200)( 79,199)( 80,198)
( 81,201)( 82,204)( 83,203)( 84,202)( 85,205)( 86,208)( 87,207)( 88,206)
( 89,209)( 90,212)( 91,211)( 92,210)( 93,213)( 94,216)( 95,215)( 96,214)
( 97,217)( 98,220)( 99,219)(100,218)(101,221)(102,224)(103,223)(104,222)
(105,225)(106,228)(107,227)(108,226)(109,229)(110,232)(111,231)(112,230)
(113,233)(114,236)(115,235)(116,234)(117,237)(118,240)(119,239)(120,238)
(121,301)(122,304)(123,303)(124,302)(125,305)(126,308)(127,307)(128,306)
(129,309)(130,312)(131,311)(132,310)(133,313)(134,316)(135,315)(136,314)
(137,317)(138,320)(139,319)(140,318)(141,321)(142,324)(143,323)(144,322)
(145,325)(146,328)(147,327)(148,326)(149,329)(150,332)(151,331)(152,330)
(153,333)(154,336)(155,335)(156,334)(157,337)(158,340)(159,339)(160,338)
(161,341)(162,344)(163,343)(164,342)(165,345)(166,348)(167,347)(168,346)
(169,349)(170,352)(171,351)(172,350)(173,353)(174,356)(175,355)(176,354)
(177,357)(178,360)(179,359)(180,358);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*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*s2*s0*s1*s0 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(360)!(  1,  3)(  2,  4)(  5, 19)(  6, 20)(  7, 17)(  8, 18)(  9, 15)
( 10, 16)( 11, 13)( 12, 14)( 21, 43)( 22, 44)( 23, 41)( 24, 42)( 25, 59)
( 26, 60)( 27, 57)( 28, 58)( 29, 55)( 30, 56)( 31, 53)( 32, 54)( 33, 51)
( 34, 52)( 35, 49)( 36, 50)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 61, 63)
( 62, 64)( 65, 79)( 66, 80)( 67, 77)( 68, 78)( 69, 75)( 70, 76)( 71, 73)
( 72, 74)( 81,103)( 82,104)( 83,101)( 84,102)( 85,119)( 86,120)( 87,117)
( 88,118)( 89,115)( 90,116)( 91,113)( 92,114)( 93,111)( 94,112)( 95,109)
( 96,110)( 97,107)( 98,108)( 99,105)(100,106)(121,123)(122,124)(125,139)
(126,140)(127,137)(128,138)(129,135)(130,136)(131,133)(132,134)(141,163)
(142,164)(143,161)(144,162)(145,179)(146,180)(147,177)(148,178)(149,175)
(150,176)(151,173)(152,174)(153,171)(154,172)(155,169)(156,170)(157,167)
(158,168)(159,165)(160,166)(181,183)(182,184)(185,199)(186,200)(187,197)
(188,198)(189,195)(190,196)(191,193)(192,194)(201,223)(202,224)(203,221)
(204,222)(205,239)(206,240)(207,237)(208,238)(209,235)(210,236)(211,233)
(212,234)(213,231)(214,232)(215,229)(216,230)(217,227)(218,228)(219,225)
(220,226)(241,243)(242,244)(245,259)(246,260)(247,257)(248,258)(249,255)
(250,256)(251,253)(252,254)(261,283)(262,284)(263,281)(264,282)(265,299)
(266,300)(267,297)(268,298)(269,295)(270,296)(271,293)(272,294)(273,291)
(274,292)(275,289)(276,290)(277,287)(278,288)(279,285)(280,286)(301,303)
(302,304)(305,319)(306,320)(307,317)(308,318)(309,315)(310,316)(311,313)
(312,314)(321,343)(322,344)(323,341)(324,342)(325,359)(326,360)(327,357)
(328,358)(329,355)(330,356)(331,353)(332,354)(333,351)(334,352)(335,349)
(336,350)(337,347)(338,348)(339,345)(340,346);
s1 := Sym(360)!(  1, 25)(  2, 26)(  3, 28)(  4, 27)(  5, 21)(  6, 22)(  7, 24)
(  8, 23)(  9, 37)( 10, 38)( 11, 40)( 12, 39)( 13, 33)( 14, 34)( 15, 36)
( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 41, 45)( 42, 46)( 43, 48)
( 44, 47)( 49, 57)( 50, 58)( 51, 60)( 52, 59)( 55, 56)( 61,145)( 62,146)
( 63,148)( 64,147)( 65,141)( 66,142)( 67,144)( 68,143)( 69,157)( 70,158)
( 71,160)( 72,159)( 73,153)( 74,154)( 75,156)( 76,155)( 77,149)( 78,150)
( 79,152)( 80,151)( 81,125)( 82,126)( 83,128)( 84,127)( 85,121)( 86,122)
( 87,124)( 88,123)( 89,137)( 90,138)( 91,140)( 92,139)( 93,133)( 94,134)
( 95,136)( 96,135)( 97,129)( 98,130)( 99,132)(100,131)(101,165)(102,166)
(103,168)(104,167)(105,161)(106,162)(107,164)(108,163)(109,177)(110,178)
(111,180)(112,179)(113,173)(114,174)(115,176)(116,175)(117,169)(118,170)
(119,172)(120,171)(181,205)(182,206)(183,208)(184,207)(185,201)(186,202)
(187,204)(188,203)(189,217)(190,218)(191,220)(192,219)(193,213)(194,214)
(195,216)(196,215)(197,209)(198,210)(199,212)(200,211)(221,225)(222,226)
(223,228)(224,227)(229,237)(230,238)(231,240)(232,239)(235,236)(241,325)
(242,326)(243,328)(244,327)(245,321)(246,322)(247,324)(248,323)(249,337)
(250,338)(251,340)(252,339)(253,333)(254,334)(255,336)(256,335)(257,329)
(258,330)(259,332)(260,331)(261,305)(262,306)(263,308)(264,307)(265,301)
(266,302)(267,304)(268,303)(269,317)(270,318)(271,320)(272,319)(273,313)
(274,314)(275,316)(276,315)(277,309)(278,310)(279,312)(280,311)(281,345)
(282,346)(283,348)(284,347)(285,341)(286,342)(287,344)(288,343)(289,357)
(290,358)(291,360)(292,359)(293,353)(294,354)(295,356)(296,355)(297,349)
(298,350)(299,352)(300,351);
s2 := Sym(360)!(  1,241)(  2,244)(  3,243)(  4,242)(  5,245)(  6,248)(  7,247)
(  8,246)(  9,249)( 10,252)( 11,251)( 12,250)( 13,253)( 14,256)( 15,255)
( 16,254)( 17,257)( 18,260)( 19,259)( 20,258)( 21,261)( 22,264)( 23,263)
( 24,262)( 25,265)( 26,268)( 27,267)( 28,266)( 29,269)( 30,272)( 31,271)
( 32,270)( 33,273)( 34,276)( 35,275)( 36,274)( 37,277)( 38,280)( 39,279)
( 40,278)( 41,281)( 42,284)( 43,283)( 44,282)( 45,285)( 46,288)( 47,287)
( 48,286)( 49,289)( 50,292)( 51,291)( 52,290)( 53,293)( 54,296)( 55,295)
( 56,294)( 57,297)( 58,300)( 59,299)( 60,298)( 61,181)( 62,184)( 63,183)
( 64,182)( 65,185)( 66,188)( 67,187)( 68,186)( 69,189)( 70,192)( 71,191)
( 72,190)( 73,193)( 74,196)( 75,195)( 76,194)( 77,197)( 78,200)( 79,199)
( 80,198)( 81,201)( 82,204)( 83,203)( 84,202)( 85,205)( 86,208)( 87,207)
( 88,206)( 89,209)( 90,212)( 91,211)( 92,210)( 93,213)( 94,216)( 95,215)
( 96,214)( 97,217)( 98,220)( 99,219)(100,218)(101,221)(102,224)(103,223)
(104,222)(105,225)(106,228)(107,227)(108,226)(109,229)(110,232)(111,231)
(112,230)(113,233)(114,236)(115,235)(116,234)(117,237)(118,240)(119,239)
(120,238)(121,301)(122,304)(123,303)(124,302)(125,305)(126,308)(127,307)
(128,306)(129,309)(130,312)(131,311)(132,310)(133,313)(134,316)(135,315)
(136,314)(137,317)(138,320)(139,319)(140,318)(141,321)(142,324)(143,323)
(144,322)(145,325)(146,328)(147,327)(148,326)(149,329)(150,332)(151,331)
(152,330)(153,333)(154,336)(155,335)(156,334)(157,337)(158,340)(159,339)
(160,338)(161,341)(162,344)(163,343)(164,342)(165,345)(166,348)(167,347)
(168,346)(169,349)(170,352)(171,351)(172,350)(173,353)(174,356)(175,355)
(176,354)(177,357)(178,360)(179,359)(180,358);
poly := sub<Sym(360)|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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*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*s2*s0*s1*s0 >;

```
References : None.
to this polytope