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# Polytope of Type {40,8}

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

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

```
References : None.
to this polytope