# Polytope of Type {5,8,2}

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

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

```

to this polytope