Polytope of Type {5,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,10}*320a
if this polytope has a name.
Group : SmallGroup(320,1582)
Rank : 3
Schlafli Type : {5,10}
Number of vertices, edges, etc : 16, 80, 32
Order of s0s1s2 : 8
Order of s0s1s2s1 : 5
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {5,10,2} of size 640
Vertex Figure Of :
   {2,5,10} of size 640
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {5,5}*160
Covers (Minimal Covers in Boldface) :
   2-fold covers : {5,10}*640, {10,10}*640b, {10,10}*640c
   4-fold covers : {5,20}*1280, {20,10}*1280a, {20,10}*1280b, {10,10}*1280a
   6-fold covers : {10,30}*1920a, {15,10}*1920
Permutation Representation (GAP) : 
s0 := (  1,161)(  2,162)(  3,181)(  4,182)(  5,187)(  6,188)(  7,175)(  8,176)
(  9,170)( 10,169)( 11,189)( 12,190)( 13,180)( 14,179)( 15,167)( 16,168)
( 17,185)( 18,186)( 19,174)( 20,173)( 21,163)( 22,164)( 23,184)( 24,183)
( 25,177)( 26,178)( 27,165)( 28,166)( 29,171)( 30,172)( 31,191)( 32,192)
( 33,289)( 34,290)( 35,309)( 36,310)( 37,315)( 38,316)( 39,303)( 40,304)
( 41,298)( 42,297)( 43,317)( 44,318)( 45,308)( 46,307)( 47,295)( 48,296)
( 49,313)( 50,314)( 51,302)( 52,301)( 53,291)( 54,292)( 55,312)( 56,311)
( 57,305)( 58,306)( 59,293)( 60,294)( 61,299)( 62,300)( 63,319)( 64,320)
( 65,257)( 66,258)( 67,277)( 68,278)( 69,283)( 70,284)( 71,271)( 72,272)
( 73,266)( 74,265)( 75,285)( 76,286)( 77,276)( 78,275)( 79,263)( 80,264)
( 81,281)( 82,282)( 83,270)( 84,269)( 85,259)( 86,260)( 87,280)( 88,279)
( 89,273)( 90,274)( 91,261)( 92,262)( 93,267)( 94,268)( 95,287)( 96,288)
( 97,225)( 98,226)( 99,245)(100,246)(101,251)(102,252)(103,239)(104,240)
(105,234)(106,233)(107,253)(108,254)(109,244)(110,243)(111,231)(112,232)
(113,249)(114,250)(115,238)(116,237)(117,227)(118,228)(119,248)(120,247)
(121,241)(122,242)(123,229)(124,230)(125,235)(126,236)(127,255)(128,256)
(129,193)(130,194)(131,213)(132,214)(133,219)(134,220)(135,207)(136,208)
(137,202)(138,201)(139,221)(140,222)(141,212)(142,211)(143,199)(144,200)
(145,217)(146,218)(147,206)(148,205)(149,195)(150,196)(151,216)(152,215)
(153,209)(154,210)(155,197)(156,198)(157,203)(158,204)(159,223)(160,224);;
s1 := (  1,193)(  2,194)(  3,223)(  4,224)(  5,198)(  6,197)(  7,219)(  8,220)
(  9,206)( 10,205)( 11,211)( 12,212)( 13,202)( 14,201)( 15,216)( 16,215)
( 17,214)( 18,213)( 19,203)( 20,204)( 21,210)( 22,209)( 23,208)( 24,207)
( 25,217)( 26,218)( 27,199)( 28,200)( 29,222)( 30,221)( 31,195)( 32,196)
( 33,161)( 34,162)( 35,191)( 36,192)( 37,166)( 38,165)( 39,187)( 40,188)
( 41,174)( 42,173)( 43,179)( 44,180)( 45,170)( 46,169)( 47,184)( 48,183)
( 49,182)( 50,181)( 51,171)( 52,172)( 53,178)( 54,177)( 55,176)( 56,175)
( 57,185)( 58,186)( 59,167)( 60,168)( 61,190)( 62,189)( 63,163)( 64,164)
( 65,289)( 66,290)( 67,319)( 68,320)( 69,294)( 70,293)( 71,315)( 72,316)
( 73,302)( 74,301)( 75,307)( 76,308)( 77,298)( 78,297)( 79,312)( 80,311)
( 81,310)( 82,309)( 83,299)( 84,300)( 85,306)( 86,305)( 87,304)( 88,303)
( 89,313)( 90,314)( 91,295)( 92,296)( 93,318)( 94,317)( 95,291)( 96,292)
( 97,257)( 98,258)( 99,287)(100,288)(101,262)(102,261)(103,283)(104,284)
(105,270)(106,269)(107,275)(108,276)(109,266)(110,265)(111,280)(112,279)
(113,278)(114,277)(115,267)(116,268)(117,274)(118,273)(119,272)(120,271)
(121,281)(122,282)(123,263)(124,264)(125,286)(126,285)(127,259)(128,260)
(129,225)(130,226)(131,255)(132,256)(133,230)(134,229)(135,251)(136,252)
(137,238)(138,237)(139,243)(140,244)(141,234)(142,233)(143,248)(144,247)
(145,246)(146,245)(147,235)(148,236)(149,242)(150,241)(151,240)(152,239)
(153,249)(154,250)(155,231)(156,232)(157,254)(158,253)(159,227)(160,228);;
s2 := (  1,192)(  2,191)(  3,171)(  4,172)(  5,165)(  6,166)(  7,178)(  8,177)
(  9,183)( 10,184)( 11,163)( 12,164)( 13,174)( 14,173)( 15,186)( 16,185)
( 17,168)( 18,167)( 19,180)( 20,179)( 21,189)( 22,190)( 23,169)( 24,170)
( 25,176)( 26,175)( 27,187)( 28,188)( 29,181)( 30,182)( 31,162)( 32,161)
( 33,320)( 34,319)( 35,299)( 36,300)( 37,293)( 38,294)( 39,306)( 40,305)
( 41,311)( 42,312)( 43,291)( 44,292)( 45,302)( 46,301)( 47,314)( 48,313)
( 49,296)( 50,295)( 51,308)( 52,307)( 53,317)( 54,318)( 55,297)( 56,298)
( 57,304)( 58,303)( 59,315)( 60,316)( 61,309)( 62,310)( 63,290)( 64,289)
( 65,288)( 66,287)( 67,267)( 68,268)( 69,261)( 70,262)( 71,274)( 72,273)
( 73,279)( 74,280)( 75,259)( 76,260)( 77,270)( 78,269)( 79,282)( 80,281)
( 81,264)( 82,263)( 83,276)( 84,275)( 85,285)( 86,286)( 87,265)( 88,266)
( 89,272)( 90,271)( 91,283)( 92,284)( 93,277)( 94,278)( 95,258)( 96,257)
( 97,256)( 98,255)( 99,235)(100,236)(101,229)(102,230)(103,242)(104,241)
(105,247)(106,248)(107,227)(108,228)(109,238)(110,237)(111,250)(112,249)
(113,232)(114,231)(115,244)(116,243)(117,253)(118,254)(119,233)(120,234)
(121,240)(122,239)(123,251)(124,252)(125,245)(126,246)(127,226)(128,225)
(129,224)(130,223)(131,203)(132,204)(133,197)(134,198)(135,210)(136,209)
(137,215)(138,216)(139,195)(140,196)(141,206)(142,205)(143,218)(144,217)
(145,200)(146,199)(147,212)(148,211)(149,221)(150,222)(151,201)(152,202)
(153,208)(154,207)(155,219)(156,220)(157,213)(158,214)(159,194)(160,193);;
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*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(320)!(  1,161)(  2,162)(  3,181)(  4,182)(  5,187)(  6,188)(  7,175)
(  8,176)(  9,170)( 10,169)( 11,189)( 12,190)( 13,180)( 14,179)( 15,167)
( 16,168)( 17,185)( 18,186)( 19,174)( 20,173)( 21,163)( 22,164)( 23,184)
( 24,183)( 25,177)( 26,178)( 27,165)( 28,166)( 29,171)( 30,172)( 31,191)
( 32,192)( 33,289)( 34,290)( 35,309)( 36,310)( 37,315)( 38,316)( 39,303)
( 40,304)( 41,298)( 42,297)( 43,317)( 44,318)( 45,308)( 46,307)( 47,295)
( 48,296)( 49,313)( 50,314)( 51,302)( 52,301)( 53,291)( 54,292)( 55,312)
( 56,311)( 57,305)( 58,306)( 59,293)( 60,294)( 61,299)( 62,300)( 63,319)
( 64,320)( 65,257)( 66,258)( 67,277)( 68,278)( 69,283)( 70,284)( 71,271)
( 72,272)( 73,266)( 74,265)( 75,285)( 76,286)( 77,276)( 78,275)( 79,263)
( 80,264)( 81,281)( 82,282)( 83,270)( 84,269)( 85,259)( 86,260)( 87,280)
( 88,279)( 89,273)( 90,274)( 91,261)( 92,262)( 93,267)( 94,268)( 95,287)
( 96,288)( 97,225)( 98,226)( 99,245)(100,246)(101,251)(102,252)(103,239)
(104,240)(105,234)(106,233)(107,253)(108,254)(109,244)(110,243)(111,231)
(112,232)(113,249)(114,250)(115,238)(116,237)(117,227)(118,228)(119,248)
(120,247)(121,241)(122,242)(123,229)(124,230)(125,235)(126,236)(127,255)
(128,256)(129,193)(130,194)(131,213)(132,214)(133,219)(134,220)(135,207)
(136,208)(137,202)(138,201)(139,221)(140,222)(141,212)(142,211)(143,199)
(144,200)(145,217)(146,218)(147,206)(148,205)(149,195)(150,196)(151,216)
(152,215)(153,209)(154,210)(155,197)(156,198)(157,203)(158,204)(159,223)
(160,224);
s1 := Sym(320)!(  1,193)(  2,194)(  3,223)(  4,224)(  5,198)(  6,197)(  7,219)
(  8,220)(  9,206)( 10,205)( 11,211)( 12,212)( 13,202)( 14,201)( 15,216)
( 16,215)( 17,214)( 18,213)( 19,203)( 20,204)( 21,210)( 22,209)( 23,208)
( 24,207)( 25,217)( 26,218)( 27,199)( 28,200)( 29,222)( 30,221)( 31,195)
( 32,196)( 33,161)( 34,162)( 35,191)( 36,192)( 37,166)( 38,165)( 39,187)
( 40,188)( 41,174)( 42,173)( 43,179)( 44,180)( 45,170)( 46,169)( 47,184)
( 48,183)( 49,182)( 50,181)( 51,171)( 52,172)( 53,178)( 54,177)( 55,176)
( 56,175)( 57,185)( 58,186)( 59,167)( 60,168)( 61,190)( 62,189)( 63,163)
( 64,164)( 65,289)( 66,290)( 67,319)( 68,320)( 69,294)( 70,293)( 71,315)
( 72,316)( 73,302)( 74,301)( 75,307)( 76,308)( 77,298)( 78,297)( 79,312)
( 80,311)( 81,310)( 82,309)( 83,299)( 84,300)( 85,306)( 86,305)( 87,304)
( 88,303)( 89,313)( 90,314)( 91,295)( 92,296)( 93,318)( 94,317)( 95,291)
( 96,292)( 97,257)( 98,258)( 99,287)(100,288)(101,262)(102,261)(103,283)
(104,284)(105,270)(106,269)(107,275)(108,276)(109,266)(110,265)(111,280)
(112,279)(113,278)(114,277)(115,267)(116,268)(117,274)(118,273)(119,272)
(120,271)(121,281)(122,282)(123,263)(124,264)(125,286)(126,285)(127,259)
(128,260)(129,225)(130,226)(131,255)(132,256)(133,230)(134,229)(135,251)
(136,252)(137,238)(138,237)(139,243)(140,244)(141,234)(142,233)(143,248)
(144,247)(145,246)(146,245)(147,235)(148,236)(149,242)(150,241)(151,240)
(152,239)(153,249)(154,250)(155,231)(156,232)(157,254)(158,253)(159,227)
(160,228);
s2 := Sym(320)!(  1,192)(  2,191)(  3,171)(  4,172)(  5,165)(  6,166)(  7,178)
(  8,177)(  9,183)( 10,184)( 11,163)( 12,164)( 13,174)( 14,173)( 15,186)
( 16,185)( 17,168)( 18,167)( 19,180)( 20,179)( 21,189)( 22,190)( 23,169)
( 24,170)( 25,176)( 26,175)( 27,187)( 28,188)( 29,181)( 30,182)( 31,162)
( 32,161)( 33,320)( 34,319)( 35,299)( 36,300)( 37,293)( 38,294)( 39,306)
( 40,305)( 41,311)( 42,312)( 43,291)( 44,292)( 45,302)( 46,301)( 47,314)
( 48,313)( 49,296)( 50,295)( 51,308)( 52,307)( 53,317)( 54,318)( 55,297)
( 56,298)( 57,304)( 58,303)( 59,315)( 60,316)( 61,309)( 62,310)( 63,290)
( 64,289)( 65,288)( 66,287)( 67,267)( 68,268)( 69,261)( 70,262)( 71,274)
( 72,273)( 73,279)( 74,280)( 75,259)( 76,260)( 77,270)( 78,269)( 79,282)
( 80,281)( 81,264)( 82,263)( 83,276)( 84,275)( 85,285)( 86,286)( 87,265)
( 88,266)( 89,272)( 90,271)( 91,283)( 92,284)( 93,277)( 94,278)( 95,258)
( 96,257)( 97,256)( 98,255)( 99,235)(100,236)(101,229)(102,230)(103,242)
(104,241)(105,247)(106,248)(107,227)(108,228)(109,238)(110,237)(111,250)
(112,249)(113,232)(114,231)(115,244)(116,243)(117,253)(118,254)(119,233)
(120,234)(121,240)(122,239)(123,251)(124,252)(125,245)(126,246)(127,226)
(128,225)(129,224)(130,223)(131,203)(132,204)(133,197)(134,198)(135,210)
(136,209)(137,215)(138,216)(139,195)(140,196)(141,206)(142,205)(143,218)
(144,217)(145,200)(146,199)(147,212)(148,211)(149,221)(150,222)(151,201)
(152,202)(153,208)(154,207)(155,219)(156,220)(157,213)(158,214)(159,194)
(160,193);
poly := sub<Sym(320)|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*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1 >;
References : None.
to this polytope