Polytope of Type {2,10,5}

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

to this polytope