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# Polytope of Type {8,20}

Atlas Canonical Name : {8,20}*640a
if this polytope has a name.
Group : SmallGroup(640,1159)
Rank : 3
Schlafli Type : {8,20}
Number of vertices, edges, etc : 16, 160, 40
Order of s0s1s2 : 40
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{8,20,2} of size 1280
Vertex Figure Of :
{2,8,20} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,20}*320, {8,20}*320a, {8,20}*320b
4-fold quotients : {4,20}*160, {8,10}*160
5-fold quotients : {8,4}*128a
8-fold quotients : {2,20}*80, {4,10}*80
10-fold quotients : {8,4}*64a, {8,4}*64b, {4,4}*64
16-fold quotients : {2,10}*40
20-fold quotients : {4,4}*32, {8,2}*32
32-fold quotients : {2,5}*20
40-fold quotients : {2,4}*16, {4,2}*16
80-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,40}*1280a, {8,20}*1280a, {8,40}*1280c, {16,20}*1280a, {16,20}*1280b
3-fold covers : {8,60}*1920a, {24,20}*1920a
Permutation Representation (GAP) :
```s0 := (  1,161)(  2,162)(  3,163)(  4,164)(  5,165)(  6,166)(  7,167)(  8,168)
(  9,169)( 10,170)( 11,171)( 12,172)( 13,173)( 14,174)( 15,175)( 16,176)
( 17,177)( 18,178)( 19,179)( 20,180)( 21,186)( 22,187)( 23,188)( 24,189)
( 25,190)( 26,181)( 27,182)( 28,183)( 29,184)( 30,185)( 31,196)( 32,197)
( 33,198)( 34,199)( 35,200)( 36,191)( 37,192)( 38,193)( 39,194)( 40,195)
( 41,211)( 42,212)( 43,213)( 44,214)( 45,215)( 46,216)( 47,217)( 48,218)
( 49,219)( 50,220)( 51,201)( 52,202)( 53,203)( 54,204)( 55,205)( 56,206)
( 57,207)( 58,208)( 59,209)( 60,210)( 61,236)( 62,237)( 63,238)( 64,239)
( 65,240)( 66,231)( 67,232)( 68,233)( 69,234)( 70,235)( 71,226)( 72,227)
( 73,228)( 74,229)( 75,230)( 76,221)( 77,222)( 78,223)( 79,224)( 80,225)
( 81,241)( 82,242)( 83,243)( 84,244)( 85,245)( 86,246)( 87,247)( 88,248)
( 89,249)( 90,250)( 91,251)( 92,252)( 93,253)( 94,254)( 95,255)( 96,256)
( 97,257)( 98,258)( 99,259)(100,260)(101,266)(102,267)(103,268)(104,269)
(105,270)(106,261)(107,262)(108,263)(109,264)(110,265)(111,276)(112,277)
(113,278)(114,279)(115,280)(116,271)(117,272)(118,273)(119,274)(120,275)
(121,291)(122,292)(123,293)(124,294)(125,295)(126,296)(127,297)(128,298)
(129,299)(130,300)(131,281)(132,282)(133,283)(134,284)(135,285)(136,286)
(137,287)(138,288)(139,289)(140,290)(141,316)(142,317)(143,318)(144,319)
(145,320)(146,311)(147,312)(148,313)(149,314)(150,315)(151,306)(152,307)
(153,308)(154,309)(155,310)(156,301)(157,302)(158,303)(159,304)(160,305);;
s1 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)( 38, 39)
( 41, 51)( 42, 55)( 43, 54)( 44, 53)( 45, 52)( 46, 56)( 47, 60)( 48, 59)
( 49, 58)( 50, 57)( 61, 71)( 62, 75)( 63, 74)( 64, 73)( 65, 72)( 66, 76)
( 67, 80)( 68, 79)( 69, 78)( 70, 77)( 81,101)( 82,105)( 83,104)( 84,103)
( 85,102)( 86,106)( 87,110)( 88,109)( 89,108)( 90,107)( 91,111)( 92,115)
( 93,114)( 94,113)( 95,112)( 96,116)( 97,120)( 98,119)( 99,118)(100,117)
(121,151)(122,155)(123,154)(124,153)(125,152)(126,156)(127,160)(128,159)
(129,158)(130,157)(131,141)(132,145)(133,144)(134,143)(135,142)(136,146)
(137,150)(138,149)(139,148)(140,147)(161,201)(162,205)(163,204)(164,203)
(165,202)(166,206)(167,210)(168,209)(169,208)(170,207)(171,211)(172,215)
(173,214)(174,213)(175,212)(176,216)(177,220)(178,219)(179,218)(180,217)
(181,221)(182,225)(183,224)(184,223)(185,222)(186,226)(187,230)(188,229)
(189,228)(190,227)(191,231)(192,235)(193,234)(194,233)(195,232)(196,236)
(197,240)(198,239)(199,238)(200,237)(241,306)(242,310)(243,309)(244,308)
(245,307)(246,301)(247,305)(248,304)(249,303)(250,302)(251,316)(252,320)
(253,319)(254,318)(255,317)(256,311)(257,315)(258,314)(259,313)(260,312)
(261,286)(262,290)(263,289)(264,288)(265,287)(266,281)(267,285)(268,284)
(269,283)(270,282)(271,296)(272,300)(273,299)(274,298)(275,297)(276,291)
(277,295)(278,294)(279,293)(280,292);;
s2 := (  1, 82)(  2, 81)(  3, 85)(  4, 84)(  5, 83)(  6, 87)(  7, 86)(  8, 90)
(  9, 89)( 10, 88)( 11, 92)( 12, 91)( 13, 95)( 14, 94)( 15, 93)( 16, 97)
( 17, 96)( 18,100)( 19, 99)( 20, 98)( 21,102)( 22,101)( 23,105)( 24,104)
( 25,103)( 26,107)( 27,106)( 28,110)( 29,109)( 30,108)( 31,112)( 32,111)
( 33,115)( 34,114)( 35,113)( 36,117)( 37,116)( 38,120)( 39,119)( 40,118)
( 41,127)( 42,126)( 43,130)( 44,129)( 45,128)( 46,122)( 47,121)( 48,125)
( 49,124)( 50,123)( 51,137)( 52,136)( 53,140)( 54,139)( 55,138)( 56,132)
( 57,131)( 58,135)( 59,134)( 60,133)( 61,147)( 62,146)( 63,150)( 64,149)
( 65,148)( 66,142)( 67,141)( 68,145)( 69,144)( 70,143)( 71,157)( 72,156)
( 73,160)( 74,159)( 75,158)( 76,152)( 77,151)( 78,155)( 79,154)( 80,153)
(161,242)(162,241)(163,245)(164,244)(165,243)(166,247)(167,246)(168,250)
(169,249)(170,248)(171,252)(172,251)(173,255)(174,254)(175,253)(176,257)
(177,256)(178,260)(179,259)(180,258)(181,262)(182,261)(183,265)(184,264)
(185,263)(186,267)(187,266)(188,270)(189,269)(190,268)(191,272)(192,271)
(193,275)(194,274)(195,273)(196,277)(197,276)(198,280)(199,279)(200,278)
(201,287)(202,286)(203,290)(204,289)(205,288)(206,282)(207,281)(208,285)
(209,284)(210,283)(211,297)(212,296)(213,300)(214,299)(215,298)(216,292)
(217,291)(218,295)(219,294)(220,293)(221,307)(222,306)(223,310)(224,309)
(225,308)(226,302)(227,301)(228,305)(229,304)(230,303)(231,317)(232,316)
(233,320)(234,319)(235,318)(236,312)(237,311)(238,315)(239,314)(240,313);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope