# Polytope of Type {8,40}

Atlas Canonical Name : {8,40}*640b
Also Known As : {8,40|2}. if this polytope has another name.
Group : SmallGroup(640,1069)
Rank : 3
Schlafli Type : {8,40}
Number of vertices, edges, etc : 8, 160, 40
Order of s0s1s2 : 40
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{8,40,2} of size 1280
Vertex Figure Of :
{2,8,40} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,40}*320a, {8,20}*320a
4-fold quotients : {4,20}*160, {2,40}*160, {8,10}*160
5-fold quotients : {8,8}*128b
8-fold quotients : {2,20}*80, {4,10}*80
10-fold quotients : {4,8}*64a, {8,4}*64a
16-fold quotients : {2,10}*40
20-fold quotients : {4,4}*32, {2,8}*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, {8,80}*1280d, {16,40}*1280d, {8,80}*1280f, {16,40}*1280f
3-fold covers : {8,120}*1920c, {24,40}*1920a
Permutation Representation (GAP) :
```s0 := (  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)(  8, 88)
(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)( 16, 96)
( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21,106)( 22,107)( 23,108)( 24,109)
( 25,110)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 31,116)( 32,117)
( 33,118)( 34,119)( 35,120)( 36,111)( 37,112)( 38,113)( 39,114)( 40,115)
( 41,126)( 42,127)( 43,128)( 44,129)( 45,130)( 46,121)( 47,122)( 48,123)
( 49,124)( 50,125)( 51,136)( 52,137)( 53,138)( 54,139)( 55,140)( 56,131)
( 57,132)( 58,133)( 59,134)( 60,135)( 61,141)( 62,142)( 63,143)( 64,144)
( 65,145)( 66,146)( 67,147)( 68,148)( 69,149)( 70,150)( 71,151)( 72,152)
( 73,153)( 74,154)( 75,155)( 76,156)( 77,157)( 78,158)( 79,159)( 80,160)
(161,241)(162,242)(163,243)(164,244)(165,245)(166,246)(167,247)(168,248)
(169,249)(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)(176,256)
(177,257)(178,258)(179,259)(180,260)(181,266)(182,267)(183,268)(184,269)
(185,270)(186,261)(187,262)(188,263)(189,264)(190,265)(191,276)(192,277)
(193,278)(194,279)(195,280)(196,271)(197,272)(198,273)(199,274)(200,275)
(201,286)(202,287)(203,288)(204,289)(205,290)(206,281)(207,282)(208,283)
(209,284)(210,285)(211,296)(212,297)(213,298)(214,299)(215,300)(216,291)
(217,292)(218,293)(219,294)(220,295)(221,301)(222,302)(223,303)(224,304)
(225,305)(226,306)(227,307)(228,308)(229,309)(230,310)(231,311)(232,312)
(233,313)(234,314)(235,315)(236,316)(237,317)(238,318)(239,319)(240,320);;
s1 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 31, 36)( 32, 40)( 33, 39)
( 34, 38)( 35, 37)( 41, 51)( 42, 55)( 43, 54)( 44, 53)( 45, 52)( 46, 56)
( 47, 60)( 48, 59)( 49, 58)( 50, 57)( 61, 76)( 62, 80)( 63, 79)( 64, 78)
( 65, 77)( 66, 71)( 67, 75)( 68, 74)( 69, 73)( 70, 72)( 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)(181,226)(182,230)(183,229)(184,228)(185,227)(186,221)
(187,225)(188,224)(189,223)(190,222)(191,236)(192,240)(193,239)(194,238)
(195,237)(196,231)(197,235)(198,234)(199,233)(200,232)(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 := (  1,242)(  2,241)(  3,245)(  4,244)(  5,243)(  6,247)(  7,246)(  8,250)
(  9,249)( 10,248)( 11,252)( 12,251)( 13,255)( 14,254)( 15,253)( 16,257)
( 17,256)( 18,260)( 19,259)( 20,258)( 21,262)( 22,261)( 23,265)( 24,264)
( 25,263)( 26,267)( 27,266)( 28,270)( 29,269)( 30,268)( 31,272)( 32,271)
( 33,275)( 34,274)( 35,273)( 36,277)( 37,276)( 38,280)( 39,279)( 40,278)
( 41,297)( 42,296)( 43,300)( 44,299)( 45,298)( 46,292)( 47,291)( 48,295)
( 49,294)( 50,293)( 51,287)( 52,286)( 53,290)( 54,289)( 55,288)( 56,282)
( 57,281)( 58,285)( 59,284)( 60,283)( 61,317)( 62,316)( 63,320)( 64,319)
( 65,318)( 66,312)( 67,311)( 68,315)( 69,314)( 70,313)( 71,307)( 72,306)
( 73,310)( 74,309)( 75,308)( 76,302)( 77,301)( 78,305)( 79,304)( 80,303)
( 81,162)( 82,161)( 83,165)( 84,164)( 85,163)( 86,167)( 87,166)( 88,170)
( 89,169)( 90,168)( 91,172)( 92,171)( 93,175)( 94,174)( 95,173)( 96,177)
( 97,176)( 98,180)( 99,179)(100,178)(101,182)(102,181)(103,185)(104,184)
(105,183)(106,187)(107,186)(108,190)(109,189)(110,188)(111,192)(112,191)
(113,195)(114,194)(115,193)(116,197)(117,196)(118,200)(119,199)(120,198)
(121,217)(122,216)(123,220)(124,219)(125,218)(126,212)(127,211)(128,215)
(129,214)(130,213)(131,207)(132,206)(133,210)(134,209)(135,208)(136,202)
(137,201)(138,205)(139,204)(140,203)(141,237)(142,236)(143,240)(144,239)
(145,238)(146,232)(147,231)(148,235)(149,234)(150,233)(151,227)(152,226)
(153,230)(154,229)(155,228)(156,222)(157,221)(158,225)(159,224)(160,223);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*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, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)
(  8, 88)(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)
( 16, 96)( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21,106)( 22,107)( 23,108)
( 24,109)( 25,110)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 31,116)
( 32,117)( 33,118)( 34,119)( 35,120)( 36,111)( 37,112)( 38,113)( 39,114)
( 40,115)( 41,126)( 42,127)( 43,128)( 44,129)( 45,130)( 46,121)( 47,122)
( 48,123)( 49,124)( 50,125)( 51,136)( 52,137)( 53,138)( 54,139)( 55,140)
( 56,131)( 57,132)( 58,133)( 59,134)( 60,135)( 61,141)( 62,142)( 63,143)
( 64,144)( 65,145)( 66,146)( 67,147)( 68,148)( 69,149)( 70,150)( 71,151)
( 72,152)( 73,153)( 74,154)( 75,155)( 76,156)( 77,157)( 78,158)( 79,159)
( 80,160)(161,241)(162,242)(163,243)(164,244)(165,245)(166,246)(167,247)
(168,248)(169,249)(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)
(176,256)(177,257)(178,258)(179,259)(180,260)(181,266)(182,267)(183,268)
(184,269)(185,270)(186,261)(187,262)(188,263)(189,264)(190,265)(191,276)
(192,277)(193,278)(194,279)(195,280)(196,271)(197,272)(198,273)(199,274)
(200,275)(201,286)(202,287)(203,288)(204,289)(205,290)(206,281)(207,282)
(208,283)(209,284)(210,285)(211,296)(212,297)(213,298)(214,299)(215,300)
(216,291)(217,292)(218,293)(219,294)(220,295)(221,301)(222,302)(223,303)
(224,304)(225,305)(226,306)(227,307)(228,308)(229,309)(230,310)(231,311)
(232,312)(233,313)(234,314)(235,315)(236,316)(237,317)(238,318)(239,319)
(240,320);
s1 := Sym(320)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 31, 36)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 41, 51)( 42, 55)( 43, 54)( 44, 53)( 45, 52)
( 46, 56)( 47, 60)( 48, 59)( 49, 58)( 50, 57)( 61, 76)( 62, 80)( 63, 79)
( 64, 78)( 65, 77)( 66, 71)( 67, 75)( 68, 74)( 69, 73)( 70, 72)( 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)(181,226)(182,230)(183,229)(184,228)(185,227)
(186,221)(187,225)(188,224)(189,223)(190,222)(191,236)(192,240)(193,239)
(194,238)(195,237)(196,231)(197,235)(198,234)(199,233)(200,232)(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)!(  1,242)(  2,241)(  3,245)(  4,244)(  5,243)(  6,247)(  7,246)
(  8,250)(  9,249)( 10,248)( 11,252)( 12,251)( 13,255)( 14,254)( 15,253)
( 16,257)( 17,256)( 18,260)( 19,259)( 20,258)( 21,262)( 22,261)( 23,265)
( 24,264)( 25,263)( 26,267)( 27,266)( 28,270)( 29,269)( 30,268)( 31,272)
( 32,271)( 33,275)( 34,274)( 35,273)( 36,277)( 37,276)( 38,280)( 39,279)
( 40,278)( 41,297)( 42,296)( 43,300)( 44,299)( 45,298)( 46,292)( 47,291)
( 48,295)( 49,294)( 50,293)( 51,287)( 52,286)( 53,290)( 54,289)( 55,288)
( 56,282)( 57,281)( 58,285)( 59,284)( 60,283)( 61,317)( 62,316)( 63,320)
( 64,319)( 65,318)( 66,312)( 67,311)( 68,315)( 69,314)( 70,313)( 71,307)
( 72,306)( 73,310)( 74,309)( 75,308)( 76,302)( 77,301)( 78,305)( 79,304)
( 80,303)( 81,162)( 82,161)( 83,165)( 84,164)( 85,163)( 86,167)( 87,166)
( 88,170)( 89,169)( 90,168)( 91,172)( 92,171)( 93,175)( 94,174)( 95,173)
( 96,177)( 97,176)( 98,180)( 99,179)(100,178)(101,182)(102,181)(103,185)
(104,184)(105,183)(106,187)(107,186)(108,190)(109,189)(110,188)(111,192)
(112,191)(113,195)(114,194)(115,193)(116,197)(117,196)(118,200)(119,199)
(120,198)(121,217)(122,216)(123,220)(124,219)(125,218)(126,212)(127,211)
(128,215)(129,214)(130,213)(131,207)(132,206)(133,210)(134,209)(135,208)
(136,202)(137,201)(138,205)(139,204)(140,203)(141,237)(142,236)(143,240)
(144,239)(145,238)(146,232)(147,231)(148,235)(149,234)(150,233)(151,227)
(152,226)(153,230)(154,229)(155,228)(156,222)(157,221)(158,225)(159,224)
(160,223);
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, s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*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