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# Polytope of Type {2,5,8}

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

```

to this polytope