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# Polytope of Type {2,8,5}

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

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

```

to this polytope