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# Polytope of Type {60,6}

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

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

```
References : None.
to this polytope