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# Polytope of Type {6,60}

Atlas Canonical Name : {6,60}*720a
if this polytope has a name.
Group : SmallGroup(720,500)
Rank : 3
Schlafli Type : {6,60}
Number of vertices, edges, etc : 6, 180, 60
Order of s0s1s2 : 60
Order of s0s1s2s1 : 6
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}*360a
3-fold quotients : {6,20}*240a
5-fold quotients : {6,12}*144c
6-fold quotients : {6,10}*120
9-fold quotients : {2,20}*80
10-fold quotients : {6,6}*72c
15-fold quotients : {6,4}*48a
18-fold quotients : {2,10}*40
20-fold quotients : {3,6}*36
30-fold quotients : {6,2}*24
36-fold quotients : {2,5}*20
45-fold quotients : {2,4}*16
60-fold quotients : {3,2}*12
90-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,120}*1440a, {12,60}*1440a
Permutation Representation (GAP) :
```s0 := (  1,271)(  2,272)(  3,273)(  4,274)(  5,275)(  6,281)(  7,282)(  8,283)
(  9,284)( 10,285)( 11,276)( 12,277)( 13,278)( 14,279)( 15,280)( 16,301)
( 17,302)( 18,303)( 19,304)( 20,305)( 21,311)( 22,312)( 23,313)( 24,314)
( 25,315)( 26,306)( 27,307)( 28,308)( 29,309)( 30,310)( 31,286)( 32,287)
( 33,288)( 34,289)( 35,290)( 36,296)( 37,297)( 38,298)( 39,299)( 40,300)
( 41,291)( 42,292)( 43,293)( 44,294)( 45,295)( 46,316)( 47,317)( 48,318)
( 49,319)( 50,320)( 51,326)( 52,327)( 53,328)( 54,329)( 55,330)( 56,321)
( 57,322)( 58,323)( 59,324)( 60,325)( 61,346)( 62,347)( 63,348)( 64,349)
( 65,350)( 66,356)( 67,357)( 68,358)( 69,359)( 70,360)( 71,351)( 72,352)
( 73,353)( 74,354)( 75,355)( 76,331)( 77,332)( 78,333)( 79,334)( 80,335)
( 81,341)( 82,342)( 83,343)( 84,344)( 85,345)( 86,336)( 87,337)( 88,338)
( 89,339)( 90,340)( 91,226)( 92,227)( 93,228)( 94,229)( 95,230)( 96,236)
( 97,237)( 98,238)( 99,239)(100,240)(101,231)(102,232)(103,233)(104,234)
(105,235)(106,256)(107,257)(108,258)(109,259)(110,260)(111,266)(112,267)
(113,268)(114,269)(115,270)(116,261)(117,262)(118,263)(119,264)(120,265)
(121,241)(122,242)(123,243)(124,244)(125,245)(126,251)(127,252)(128,253)
(129,254)(130,255)(131,246)(132,247)(133,248)(134,249)(135,250)(136,181)
(137,182)(138,183)(139,184)(140,185)(141,191)(142,192)(143,193)(144,194)
(145,195)(146,186)(147,187)(148,188)(149,189)(150,190)(151,211)(152,212)
(153,213)(154,214)(155,215)(156,221)(157,222)(158,223)(159,224)(160,225)
(161,216)(162,217)(163,218)(164,219)(165,220)(166,196)(167,197)(168,198)
(169,199)(170,200)(171,206)(172,207)(173,208)(174,209)(175,210)(176,201)
(177,202)(178,203)(179,204)(180,205);;
s1 := (  1,291)(  2,295)(  3,294)(  4,293)(  5,292)(  6,286)(  7,290)(  8,289)
(  9,288)( 10,287)( 11,296)( 12,300)( 13,299)( 14,298)( 15,297)( 16,276)
( 17,280)( 18,279)( 19,278)( 20,277)( 21,271)( 22,275)( 23,274)( 24,273)
( 25,272)( 26,281)( 27,285)( 28,284)( 29,283)( 30,282)( 31,306)( 32,310)
( 33,309)( 34,308)( 35,307)( 36,301)( 37,305)( 38,304)( 39,303)( 40,302)
( 41,311)( 42,315)( 43,314)( 44,313)( 45,312)( 46,336)( 47,340)( 48,339)
( 49,338)( 50,337)( 51,331)( 52,335)( 53,334)( 54,333)( 55,332)( 56,341)
( 57,345)( 58,344)( 59,343)( 60,342)( 61,321)( 62,325)( 63,324)( 64,323)
( 65,322)( 66,316)( 67,320)( 68,319)( 69,318)( 70,317)( 71,326)( 72,330)
( 73,329)( 74,328)( 75,327)( 76,351)( 77,355)( 78,354)( 79,353)( 80,352)
( 81,346)( 82,350)( 83,349)( 84,348)( 85,347)( 86,356)( 87,360)( 88,359)
( 89,358)( 90,357)( 91,201)( 92,205)( 93,204)( 94,203)( 95,202)( 96,196)
( 97,200)( 98,199)( 99,198)(100,197)(101,206)(102,210)(103,209)(104,208)
(105,207)(106,186)(107,190)(108,189)(109,188)(110,187)(111,181)(112,185)
(113,184)(114,183)(115,182)(116,191)(117,195)(118,194)(119,193)(120,192)
(121,216)(122,220)(123,219)(124,218)(125,217)(126,211)(127,215)(128,214)
(129,213)(130,212)(131,221)(132,225)(133,224)(134,223)(135,222)(136,246)
(137,250)(138,249)(139,248)(140,247)(141,241)(142,245)(143,244)(144,243)
(145,242)(146,251)(147,255)(148,254)(149,253)(150,252)(151,231)(152,235)
(153,234)(154,233)(155,232)(156,226)(157,230)(158,229)(159,228)(160,227)
(161,236)(162,240)(163,239)(164,238)(165,237)(166,261)(167,265)(168,264)
(169,263)(170,262)(171,256)(172,260)(173,259)(174,258)(175,257)(176,266)
(177,270)(178,269)(179,268)(180,267);;
s2 := (  1,182)(  2,181)(  3,185)(  4,184)(  5,183)(  6,192)(  7,191)(  8,195)
(  9,194)( 10,193)( 11,187)( 12,186)( 13,190)( 14,189)( 15,188)( 16,197)
( 17,196)( 18,200)( 19,199)( 20,198)( 21,207)( 22,206)( 23,210)( 24,209)
( 25,208)( 26,202)( 27,201)( 28,205)( 29,204)( 30,203)( 31,212)( 32,211)
( 33,215)( 34,214)( 35,213)( 36,222)( 37,221)( 38,225)( 39,224)( 40,223)
( 41,217)( 42,216)( 43,220)( 44,219)( 45,218)( 46,227)( 47,226)( 48,230)
( 49,229)( 50,228)( 51,237)( 52,236)( 53,240)( 54,239)( 55,238)( 56,232)
( 57,231)( 58,235)( 59,234)( 60,233)( 61,242)( 62,241)( 63,245)( 64,244)
( 65,243)( 66,252)( 67,251)( 68,255)( 69,254)( 70,253)( 71,247)( 72,246)
( 73,250)( 74,249)( 75,248)( 76,257)( 77,256)( 78,260)( 79,259)( 80,258)
( 81,267)( 82,266)( 83,270)( 84,269)( 85,268)( 86,262)( 87,261)( 88,265)
( 89,264)( 90,263)( 91,317)( 92,316)( 93,320)( 94,319)( 95,318)( 96,327)
( 97,326)( 98,330)( 99,329)(100,328)(101,322)(102,321)(103,325)(104,324)
(105,323)(106,332)(107,331)(108,335)(109,334)(110,333)(111,342)(112,341)
(113,345)(114,344)(115,343)(116,337)(117,336)(118,340)(119,339)(120,338)
(121,347)(122,346)(123,350)(124,349)(125,348)(126,357)(127,356)(128,360)
(129,359)(130,358)(131,352)(132,351)(133,355)(134,354)(135,353)(136,272)
(137,271)(138,275)(139,274)(140,273)(141,282)(142,281)(143,285)(144,284)
(145,283)(146,277)(147,276)(148,280)(149,279)(150,278)(151,287)(152,286)
(153,290)(154,289)(155,288)(156,297)(157,296)(158,300)(159,299)(160,298)
(161,292)(162,291)(163,295)(164,294)(165,293)(166,302)(167,301)(168,305)
(169,304)(170,303)(171,312)(172,311)(173,315)(174,314)(175,313)(176,307)
(177,306)(178,310)(179,309)(180,308);;
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*s1*s0*s1*s2*s0*s1*s2*s1*s0*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*s0*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope