Polytope of Type {2,5,10}

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

to this polytope