Polytope of Type {2,5,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,5,10}*640b
if this polytope has a name.
Group : SmallGroup(640,21537)
Rank : 4
Schlafli Type : {2,5,10}
Number of vertices, edges, etc : 2, 16, 80, 32
Order of s0s1s2s3 : 4
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}*1280a, {2,5,20}*1280a, {2,5,20}*1280b, {2,10,10}*1280d
   3-fold covers : {2,15,10}*1920
Permutation Representation (GAP) : 
s0 := (1,2);;
s1 := (  3,163)(  4,173)(  5,176)(  6,170)(  7,167)(  8,177)(  9,172)( 10,166)
( 11,175)( 12,169)( 13,164)( 14,174)( 15,171)( 16,165)( 17,168)( 18,178)
( 19,227)( 20,237)( 21,240)( 22,234)( 23,231)( 24,241)( 25,236)( 26,230)
( 27,239)( 28,233)( 29,228)( 30,238)( 31,235)( 32,229)( 33,232)( 34,242)
( 35,211)( 36,221)( 37,224)( 38,218)( 39,215)( 40,225)( 41,220)( 42,214)
( 43,223)( 44,217)( 45,212)( 46,222)( 47,219)( 48,213)( 49,216)( 50,226)
( 51,195)( 52,205)( 53,208)( 54,202)( 55,199)( 56,209)( 57,204)( 58,198)
( 59,207)( 60,201)( 61,196)( 62,206)( 63,203)( 64,197)( 65,200)( 66,210)
( 67,179)( 68,189)( 69,192)( 70,186)( 71,183)( 72,193)( 73,188)( 74,182)
( 75,191)( 76,185)( 77,180)( 78,190)( 79,187)( 80,181)( 81,184)( 82,194)
( 83,243)( 84,253)( 85,256)( 86,250)( 87,247)( 88,257)( 89,252)( 90,246)
( 91,255)( 92,249)( 93,244)( 94,254)( 95,251)( 96,245)( 97,248)( 98,258)
( 99,307)(100,317)(101,320)(102,314)(103,311)(104,321)(105,316)(106,310)
(107,319)(108,313)(109,308)(110,318)(111,315)(112,309)(113,312)(114,322)
(115,291)(116,301)(117,304)(118,298)(119,295)(120,305)(121,300)(122,294)
(123,303)(124,297)(125,292)(126,302)(127,299)(128,293)(129,296)(130,306)
(131,275)(132,285)(133,288)(134,282)(135,279)(136,289)(137,284)(138,278)
(139,287)(140,281)(141,276)(142,286)(143,283)(144,277)(145,280)(146,290)
(147,259)(148,269)(149,272)(150,266)(151,263)(152,273)(153,268)(154,262)
(155,271)(156,265)(157,260)(158,270)(159,267)(160,261)(161,264)(162,274);;
s2 := (  3,179)(  4,194)(  5,181)(  6,192)(  7,185)(  8,188)(  9,183)( 10,190)
( 11,189)( 12,184)( 13,187)( 14,186)( 15,191)( 16,182)( 17,193)( 18,180)
( 19,163)( 20,178)( 21,165)( 22,176)( 23,169)( 24,172)( 25,167)( 26,174)
( 27,173)( 28,168)( 29,171)( 30,170)( 31,175)( 32,166)( 33,177)( 34,164)
( 35,227)( 36,242)( 37,229)( 38,240)( 39,233)( 40,236)( 41,231)( 42,238)
( 43,237)( 44,232)( 45,235)( 46,234)( 47,239)( 48,230)( 49,241)( 50,228)
( 51,211)( 52,226)( 53,213)( 54,224)( 55,217)( 56,220)( 57,215)( 58,222)
( 59,221)( 60,216)( 61,219)( 62,218)( 63,223)( 64,214)( 65,225)( 66,212)
( 67,195)( 68,210)( 69,197)( 70,208)( 71,201)( 72,204)( 73,199)( 74,206)
( 75,205)( 76,200)( 77,203)( 78,202)( 79,207)( 80,198)( 81,209)( 82,196)
( 83,259)( 84,274)( 85,261)( 86,272)( 87,265)( 88,268)( 89,263)( 90,270)
( 91,269)( 92,264)( 93,267)( 94,266)( 95,271)( 96,262)( 97,273)( 98,260)
( 99,243)(100,258)(101,245)(102,256)(103,249)(104,252)(105,247)(106,254)
(107,253)(108,248)(109,251)(110,250)(111,255)(112,246)(113,257)(114,244)
(115,307)(116,322)(117,309)(118,320)(119,313)(120,316)(121,311)(122,318)
(123,317)(124,312)(125,315)(126,314)(127,319)(128,310)(129,321)(130,308)
(131,291)(132,306)(133,293)(134,304)(135,297)(136,300)(137,295)(138,302)
(139,301)(140,296)(141,299)(142,298)(143,303)(144,294)(145,305)(146,292)
(147,275)(148,290)(149,277)(150,288)(151,281)(152,284)(153,279)(154,286)
(155,285)(156,280)(157,283)(158,282)(159,287)(160,278)(161,289)(162,276);;
s3 := (  3,247)(  4,257)(  5,252)(  6,246)(  7,243)(  8,253)(  9,256)( 10,250)
( 11,251)( 12,245)( 13,248)( 14,258)( 15,255)( 16,249)( 17,244)( 18,254)
( 19,311)( 20,321)( 21,316)( 22,310)( 23,307)( 24,317)( 25,320)( 26,314)
( 27,315)( 28,309)( 29,312)( 30,322)( 31,319)( 32,313)( 33,308)( 34,318)
( 35,295)( 36,305)( 37,300)( 38,294)( 39,291)( 40,301)( 41,304)( 42,298)
( 43,299)( 44,293)( 45,296)( 46,306)( 47,303)( 48,297)( 49,292)( 50,302)
( 51,279)( 52,289)( 53,284)( 54,278)( 55,275)( 56,285)( 57,288)( 58,282)
( 59,283)( 60,277)( 61,280)( 62,290)( 63,287)( 64,281)( 65,276)( 66,286)
( 67,263)( 68,273)( 69,268)( 70,262)( 71,259)( 72,269)( 73,272)( 74,266)
( 75,267)( 76,261)( 77,264)( 78,274)( 79,271)( 80,265)( 81,260)( 82,270)
( 83,167)( 84,177)( 85,172)( 86,166)( 87,163)( 88,173)( 89,176)( 90,170)
( 91,171)( 92,165)( 93,168)( 94,178)( 95,175)( 96,169)( 97,164)( 98,174)
( 99,231)(100,241)(101,236)(102,230)(103,227)(104,237)(105,240)(106,234)
(107,235)(108,229)(109,232)(110,242)(111,239)(112,233)(113,228)(114,238)
(115,215)(116,225)(117,220)(118,214)(119,211)(120,221)(121,224)(122,218)
(123,219)(124,213)(125,216)(126,226)(127,223)(128,217)(129,212)(130,222)
(131,199)(132,209)(133,204)(134,198)(135,195)(136,205)(137,208)(138,202)
(139,203)(140,197)(141,200)(142,210)(143,207)(144,201)(145,196)(146,206)
(147,183)(148,193)(149,188)(150,182)(151,179)(152,189)(153,192)(154,186)
(155,187)(156,181)(157,184)(158,194)(159,191)(160,185)(161,180)(162,190);;
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*s3*s1*s2*s3*s1*s2, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(322)!(1,2);
s1 := Sym(322)!(  3,163)(  4,173)(  5,176)(  6,170)(  7,167)(  8,177)(  9,172)
( 10,166)( 11,175)( 12,169)( 13,164)( 14,174)( 15,171)( 16,165)( 17,168)
( 18,178)( 19,227)( 20,237)( 21,240)( 22,234)( 23,231)( 24,241)( 25,236)
( 26,230)( 27,239)( 28,233)( 29,228)( 30,238)( 31,235)( 32,229)( 33,232)
( 34,242)( 35,211)( 36,221)( 37,224)( 38,218)( 39,215)( 40,225)( 41,220)
( 42,214)( 43,223)( 44,217)( 45,212)( 46,222)( 47,219)( 48,213)( 49,216)
( 50,226)( 51,195)( 52,205)( 53,208)( 54,202)( 55,199)( 56,209)( 57,204)
( 58,198)( 59,207)( 60,201)( 61,196)( 62,206)( 63,203)( 64,197)( 65,200)
( 66,210)( 67,179)( 68,189)( 69,192)( 70,186)( 71,183)( 72,193)( 73,188)
( 74,182)( 75,191)( 76,185)( 77,180)( 78,190)( 79,187)( 80,181)( 81,184)
( 82,194)( 83,243)( 84,253)( 85,256)( 86,250)( 87,247)( 88,257)( 89,252)
( 90,246)( 91,255)( 92,249)( 93,244)( 94,254)( 95,251)( 96,245)( 97,248)
( 98,258)( 99,307)(100,317)(101,320)(102,314)(103,311)(104,321)(105,316)
(106,310)(107,319)(108,313)(109,308)(110,318)(111,315)(112,309)(113,312)
(114,322)(115,291)(116,301)(117,304)(118,298)(119,295)(120,305)(121,300)
(122,294)(123,303)(124,297)(125,292)(126,302)(127,299)(128,293)(129,296)
(130,306)(131,275)(132,285)(133,288)(134,282)(135,279)(136,289)(137,284)
(138,278)(139,287)(140,281)(141,276)(142,286)(143,283)(144,277)(145,280)
(146,290)(147,259)(148,269)(149,272)(150,266)(151,263)(152,273)(153,268)
(154,262)(155,271)(156,265)(157,260)(158,270)(159,267)(160,261)(161,264)
(162,274);
s2 := Sym(322)!(  3,179)(  4,194)(  5,181)(  6,192)(  7,185)(  8,188)(  9,183)
( 10,190)( 11,189)( 12,184)( 13,187)( 14,186)( 15,191)( 16,182)( 17,193)
( 18,180)( 19,163)( 20,178)( 21,165)( 22,176)( 23,169)( 24,172)( 25,167)
( 26,174)( 27,173)( 28,168)( 29,171)( 30,170)( 31,175)( 32,166)( 33,177)
( 34,164)( 35,227)( 36,242)( 37,229)( 38,240)( 39,233)( 40,236)( 41,231)
( 42,238)( 43,237)( 44,232)( 45,235)( 46,234)( 47,239)( 48,230)( 49,241)
( 50,228)( 51,211)( 52,226)( 53,213)( 54,224)( 55,217)( 56,220)( 57,215)
( 58,222)( 59,221)( 60,216)( 61,219)( 62,218)( 63,223)( 64,214)( 65,225)
( 66,212)( 67,195)( 68,210)( 69,197)( 70,208)( 71,201)( 72,204)( 73,199)
( 74,206)( 75,205)( 76,200)( 77,203)( 78,202)( 79,207)( 80,198)( 81,209)
( 82,196)( 83,259)( 84,274)( 85,261)( 86,272)( 87,265)( 88,268)( 89,263)
( 90,270)( 91,269)( 92,264)( 93,267)( 94,266)( 95,271)( 96,262)( 97,273)
( 98,260)( 99,243)(100,258)(101,245)(102,256)(103,249)(104,252)(105,247)
(106,254)(107,253)(108,248)(109,251)(110,250)(111,255)(112,246)(113,257)
(114,244)(115,307)(116,322)(117,309)(118,320)(119,313)(120,316)(121,311)
(122,318)(123,317)(124,312)(125,315)(126,314)(127,319)(128,310)(129,321)
(130,308)(131,291)(132,306)(133,293)(134,304)(135,297)(136,300)(137,295)
(138,302)(139,301)(140,296)(141,299)(142,298)(143,303)(144,294)(145,305)
(146,292)(147,275)(148,290)(149,277)(150,288)(151,281)(152,284)(153,279)
(154,286)(155,285)(156,280)(157,283)(158,282)(159,287)(160,278)(161,289)
(162,276);
s3 := Sym(322)!(  3,247)(  4,257)(  5,252)(  6,246)(  7,243)(  8,253)(  9,256)
( 10,250)( 11,251)( 12,245)( 13,248)( 14,258)( 15,255)( 16,249)( 17,244)
( 18,254)( 19,311)( 20,321)( 21,316)( 22,310)( 23,307)( 24,317)( 25,320)
( 26,314)( 27,315)( 28,309)( 29,312)( 30,322)( 31,319)( 32,313)( 33,308)
( 34,318)( 35,295)( 36,305)( 37,300)( 38,294)( 39,291)( 40,301)( 41,304)
( 42,298)( 43,299)( 44,293)( 45,296)( 46,306)( 47,303)( 48,297)( 49,292)
( 50,302)( 51,279)( 52,289)( 53,284)( 54,278)( 55,275)( 56,285)( 57,288)
( 58,282)( 59,283)( 60,277)( 61,280)( 62,290)( 63,287)( 64,281)( 65,276)
( 66,286)( 67,263)( 68,273)( 69,268)( 70,262)( 71,259)( 72,269)( 73,272)
( 74,266)( 75,267)( 76,261)( 77,264)( 78,274)( 79,271)( 80,265)( 81,260)
( 82,270)( 83,167)( 84,177)( 85,172)( 86,166)( 87,163)( 88,173)( 89,176)
( 90,170)( 91,171)( 92,165)( 93,168)( 94,178)( 95,175)( 96,169)( 97,164)
( 98,174)( 99,231)(100,241)(101,236)(102,230)(103,227)(104,237)(105,240)
(106,234)(107,235)(108,229)(109,232)(110,242)(111,239)(112,233)(113,228)
(114,238)(115,215)(116,225)(117,220)(118,214)(119,211)(120,221)(121,224)
(122,218)(123,219)(124,213)(125,216)(126,226)(127,223)(128,217)(129,212)
(130,222)(131,199)(132,209)(133,204)(134,198)(135,195)(136,205)(137,208)
(138,202)(139,203)(140,197)(141,200)(142,210)(143,207)(144,201)(145,196)
(146,206)(147,183)(148,193)(149,188)(150,182)(151,179)(152,189)(153,192)
(154,186)(155,187)(156,181)(157,184)(158,194)(159,191)(160,185)(161,180)
(162,190);
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*s3*s1*s2*s3*s1*s2, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 >;

to this polytope