Polytope of Type {2,5,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,5,5}*640
if this polytope has a name.
Group : SmallGroup(640,21461)
Rank : 4
Schlafli Type : {2,5,5}
Number of vertices, edges, etc : 2, 32, 80, 32
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,5,5,2} of size 1280
Vertex Figure Of :
   {2,2,5,5} of size 1280
   {3,2,5,5} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,5,5}*320
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,5,10}*1280, {2,10,5}*1280
Permutation Representation (GAP) : 
s0 := (1,2);;
s1 := (  3,163)(  4,164)(  5,183)(  6,184)(  7,189)(  8,190)(  9,177)( 10,178)
( 11,172)( 12,171)( 13,191)( 14,192)( 15,182)( 16,181)( 17,169)( 18,170)
( 19,187)( 20,188)( 21,176)( 22,175)( 23,165)( 24,166)( 25,186)( 26,185)
( 27,179)( 28,180)( 29,167)( 30,168)( 31,173)( 32,174)( 33,193)( 34,194)
( 35,291)( 36,292)( 37,311)( 38,312)( 39,317)( 40,318)( 41,305)( 42,306)
( 43,300)( 44,299)( 45,319)( 46,320)( 47,310)( 48,309)( 49,297)( 50,298)
( 51,315)( 52,316)( 53,304)( 54,303)( 55,293)( 56,294)( 57,314)( 58,313)
( 59,307)( 60,308)( 61,295)( 62,296)( 63,301)( 64,302)( 65,321)( 66,322)
( 67,259)( 68,260)( 69,279)( 70,280)( 71,285)( 72,286)( 73,273)( 74,274)
( 75,268)( 76,267)( 77,287)( 78,288)( 79,278)( 80,277)( 81,265)( 82,266)
( 83,283)( 84,284)( 85,272)( 86,271)( 87,261)( 88,262)( 89,282)( 90,281)
( 91,275)( 92,276)( 93,263)( 94,264)( 95,269)( 96,270)( 97,289)( 98,290)
( 99,227)(100,228)(101,247)(102,248)(103,253)(104,254)(105,241)(106,242)
(107,236)(108,235)(109,255)(110,256)(111,246)(112,245)(113,233)(114,234)
(115,251)(116,252)(117,240)(118,239)(119,229)(120,230)(121,250)(122,249)
(123,243)(124,244)(125,231)(126,232)(127,237)(128,238)(129,257)(130,258)
(131,195)(132,196)(133,215)(134,216)(135,221)(136,222)(137,209)(138,210)
(139,204)(140,203)(141,223)(142,224)(143,214)(144,213)(145,201)(146,202)
(147,219)(148,220)(149,208)(150,207)(151,197)(152,198)(153,218)(154,217)
(155,211)(156,212)(157,199)(158,200)(159,205)(160,206)(161,225)(162,226);;
s2 := (  3,195)(  4,196)(  5,225)(  6,226)(  7,200)(  8,199)(  9,221)( 10,222)
( 11,208)( 12,207)( 13,213)( 14,214)( 15,204)( 16,203)( 17,218)( 18,217)
( 19,216)( 20,215)( 21,205)( 22,206)( 23,212)( 24,211)( 25,210)( 26,209)
( 27,219)( 28,220)( 29,201)( 30,202)( 31,224)( 32,223)( 33,197)( 34,198)
( 35,163)( 36,164)( 37,193)( 38,194)( 39,168)( 40,167)( 41,189)( 42,190)
( 43,176)( 44,175)( 45,181)( 46,182)( 47,172)( 48,171)( 49,186)( 50,185)
( 51,184)( 52,183)( 53,173)( 54,174)( 55,180)( 56,179)( 57,178)( 58,177)
( 59,187)( 60,188)( 61,169)( 62,170)( 63,192)( 64,191)( 65,165)( 66,166)
( 67,291)( 68,292)( 69,321)( 70,322)( 71,296)( 72,295)( 73,317)( 74,318)
( 75,304)( 76,303)( 77,309)( 78,310)( 79,300)( 80,299)( 81,314)( 82,313)
( 83,312)( 84,311)( 85,301)( 86,302)( 87,308)( 88,307)( 89,306)( 90,305)
( 91,315)( 92,316)( 93,297)( 94,298)( 95,320)( 96,319)( 97,293)( 98,294)
( 99,259)(100,260)(101,289)(102,290)(103,264)(104,263)(105,285)(106,286)
(107,272)(108,271)(109,277)(110,278)(111,268)(112,267)(113,282)(114,281)
(115,280)(116,279)(117,269)(118,270)(119,276)(120,275)(121,274)(122,273)
(123,283)(124,284)(125,265)(126,266)(127,288)(128,287)(129,261)(130,262)
(131,227)(132,228)(133,257)(134,258)(135,232)(136,231)(137,253)(138,254)
(139,240)(140,239)(141,245)(142,246)(143,236)(144,235)(145,250)(146,249)
(147,248)(148,247)(149,237)(150,238)(151,244)(152,243)(153,242)(154,241)
(155,251)(156,252)(157,233)(158,234)(159,256)(160,255)(161,229)(162,230);;
s3 := (  3,193)(  4,194)(  5,174)(  6,173)(  7,168)(  8,167)(  9,179)( 10,180)
( 11,186)( 12,185)( 13,166)( 14,165)( 15,175)( 16,176)( 17,187)( 18,188)
( 19,169)( 20,170)( 21,181)( 22,182)( 23,192)( 24,191)( 25,172)( 26,171)
( 27,177)( 28,178)( 29,190)( 30,189)( 31,184)( 32,183)( 33,163)( 34,164)
( 35,321)( 36,322)( 37,302)( 38,301)( 39,296)( 40,295)( 41,307)( 42,308)
( 43,314)( 44,313)( 45,294)( 46,293)( 47,303)( 48,304)( 49,315)( 50,316)
( 51,297)( 52,298)( 53,309)( 54,310)( 55,320)( 56,319)( 57,300)( 58,299)
( 59,305)( 60,306)( 61,318)( 62,317)( 63,312)( 64,311)( 65,291)( 66,292)
( 67,289)( 68,290)( 69,270)( 70,269)( 71,264)( 72,263)( 73,275)( 74,276)
( 75,282)( 76,281)( 77,262)( 78,261)( 79,271)( 80,272)( 81,283)( 82,284)
( 83,265)( 84,266)( 85,277)( 86,278)( 87,288)( 88,287)( 89,268)( 90,267)
( 91,273)( 92,274)( 93,286)( 94,285)( 95,280)( 96,279)( 97,259)( 98,260)
( 99,257)(100,258)(101,238)(102,237)(103,232)(104,231)(105,243)(106,244)
(107,250)(108,249)(109,230)(110,229)(111,239)(112,240)(113,251)(114,252)
(115,233)(116,234)(117,245)(118,246)(119,256)(120,255)(121,236)(122,235)
(123,241)(124,242)(125,254)(126,253)(127,248)(128,247)(129,227)(130,228)
(131,225)(132,226)(133,206)(134,205)(135,200)(136,199)(137,211)(138,212)
(139,218)(140,217)(141,198)(142,197)(143,207)(144,208)(145,219)(146,220)
(147,201)(148,202)(149,213)(150,214)(151,224)(152,223)(153,204)(154,203)
(155,209)(156,210)(157,222)(158,221)(159,216)(160,215)(161,195)(162,196);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(322)!(1,2);
s1 := Sym(322)!(  3,163)(  4,164)(  5,183)(  6,184)(  7,189)(  8,190)(  9,177)
( 10,178)( 11,172)( 12,171)( 13,191)( 14,192)( 15,182)( 16,181)( 17,169)
( 18,170)( 19,187)( 20,188)( 21,176)( 22,175)( 23,165)( 24,166)( 25,186)
( 26,185)( 27,179)( 28,180)( 29,167)( 30,168)( 31,173)( 32,174)( 33,193)
( 34,194)( 35,291)( 36,292)( 37,311)( 38,312)( 39,317)( 40,318)( 41,305)
( 42,306)( 43,300)( 44,299)( 45,319)( 46,320)( 47,310)( 48,309)( 49,297)
( 50,298)( 51,315)( 52,316)( 53,304)( 54,303)( 55,293)( 56,294)( 57,314)
( 58,313)( 59,307)( 60,308)( 61,295)( 62,296)( 63,301)( 64,302)( 65,321)
( 66,322)( 67,259)( 68,260)( 69,279)( 70,280)( 71,285)( 72,286)( 73,273)
( 74,274)( 75,268)( 76,267)( 77,287)( 78,288)( 79,278)( 80,277)( 81,265)
( 82,266)( 83,283)( 84,284)( 85,272)( 86,271)( 87,261)( 88,262)( 89,282)
( 90,281)( 91,275)( 92,276)( 93,263)( 94,264)( 95,269)( 96,270)( 97,289)
( 98,290)( 99,227)(100,228)(101,247)(102,248)(103,253)(104,254)(105,241)
(106,242)(107,236)(108,235)(109,255)(110,256)(111,246)(112,245)(113,233)
(114,234)(115,251)(116,252)(117,240)(118,239)(119,229)(120,230)(121,250)
(122,249)(123,243)(124,244)(125,231)(126,232)(127,237)(128,238)(129,257)
(130,258)(131,195)(132,196)(133,215)(134,216)(135,221)(136,222)(137,209)
(138,210)(139,204)(140,203)(141,223)(142,224)(143,214)(144,213)(145,201)
(146,202)(147,219)(148,220)(149,208)(150,207)(151,197)(152,198)(153,218)
(154,217)(155,211)(156,212)(157,199)(158,200)(159,205)(160,206)(161,225)
(162,226);
s2 := Sym(322)!(  3,195)(  4,196)(  5,225)(  6,226)(  7,200)(  8,199)(  9,221)
( 10,222)( 11,208)( 12,207)( 13,213)( 14,214)( 15,204)( 16,203)( 17,218)
( 18,217)( 19,216)( 20,215)( 21,205)( 22,206)( 23,212)( 24,211)( 25,210)
( 26,209)( 27,219)( 28,220)( 29,201)( 30,202)( 31,224)( 32,223)( 33,197)
( 34,198)( 35,163)( 36,164)( 37,193)( 38,194)( 39,168)( 40,167)( 41,189)
( 42,190)( 43,176)( 44,175)( 45,181)( 46,182)( 47,172)( 48,171)( 49,186)
( 50,185)( 51,184)( 52,183)( 53,173)( 54,174)( 55,180)( 56,179)( 57,178)
( 58,177)( 59,187)( 60,188)( 61,169)( 62,170)( 63,192)( 64,191)( 65,165)
( 66,166)( 67,291)( 68,292)( 69,321)( 70,322)( 71,296)( 72,295)( 73,317)
( 74,318)( 75,304)( 76,303)( 77,309)( 78,310)( 79,300)( 80,299)( 81,314)
( 82,313)( 83,312)( 84,311)( 85,301)( 86,302)( 87,308)( 88,307)( 89,306)
( 90,305)( 91,315)( 92,316)( 93,297)( 94,298)( 95,320)( 96,319)( 97,293)
( 98,294)( 99,259)(100,260)(101,289)(102,290)(103,264)(104,263)(105,285)
(106,286)(107,272)(108,271)(109,277)(110,278)(111,268)(112,267)(113,282)
(114,281)(115,280)(116,279)(117,269)(118,270)(119,276)(120,275)(121,274)
(122,273)(123,283)(124,284)(125,265)(126,266)(127,288)(128,287)(129,261)
(130,262)(131,227)(132,228)(133,257)(134,258)(135,232)(136,231)(137,253)
(138,254)(139,240)(140,239)(141,245)(142,246)(143,236)(144,235)(145,250)
(146,249)(147,248)(148,247)(149,237)(150,238)(151,244)(152,243)(153,242)
(154,241)(155,251)(156,252)(157,233)(158,234)(159,256)(160,255)(161,229)
(162,230);
s3 := Sym(322)!(  3,193)(  4,194)(  5,174)(  6,173)(  7,168)(  8,167)(  9,179)
( 10,180)( 11,186)( 12,185)( 13,166)( 14,165)( 15,175)( 16,176)( 17,187)
( 18,188)( 19,169)( 20,170)( 21,181)( 22,182)( 23,192)( 24,191)( 25,172)
( 26,171)( 27,177)( 28,178)( 29,190)( 30,189)( 31,184)( 32,183)( 33,163)
( 34,164)( 35,321)( 36,322)( 37,302)( 38,301)( 39,296)( 40,295)( 41,307)
( 42,308)( 43,314)( 44,313)( 45,294)( 46,293)( 47,303)( 48,304)( 49,315)
( 50,316)( 51,297)( 52,298)( 53,309)( 54,310)( 55,320)( 56,319)( 57,300)
( 58,299)( 59,305)( 60,306)( 61,318)( 62,317)( 63,312)( 64,311)( 65,291)
( 66,292)( 67,289)( 68,290)( 69,270)( 70,269)( 71,264)( 72,263)( 73,275)
( 74,276)( 75,282)( 76,281)( 77,262)( 78,261)( 79,271)( 80,272)( 81,283)
( 82,284)( 83,265)( 84,266)( 85,277)( 86,278)( 87,288)( 88,287)( 89,268)
( 90,267)( 91,273)( 92,274)( 93,286)( 94,285)( 95,280)( 96,279)( 97,259)
( 98,260)( 99,257)(100,258)(101,238)(102,237)(103,232)(104,231)(105,243)
(106,244)(107,250)(108,249)(109,230)(110,229)(111,239)(112,240)(113,251)
(114,252)(115,233)(116,234)(117,245)(118,246)(119,256)(120,255)(121,236)
(122,235)(123,241)(124,242)(125,254)(126,253)(127,248)(128,247)(129,227)
(130,228)(131,225)(132,226)(133,206)(134,205)(135,200)(136,199)(137,211)
(138,212)(139,218)(140,217)(141,198)(142,197)(143,207)(144,208)(145,219)
(146,220)(147,201)(148,202)(149,213)(150,214)(151,224)(152,223)(153,204)
(154,203)(155,209)(156,210)(157,222)(158,221)(159,216)(160,215)(161,195)
(162,196);
poly := sub<Sym(322)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2 >;

to this polytope