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# Polytope of Type {8,40}

Atlas Canonical Name : {8,40}*640d
if this polytope has a name.
Group : SmallGroup(640,1265)
Rank : 3
Schlafli Type : {8,40}
Number of vertices, edges, etc : 8, 160, 40
Order of s0s1s2 : 40
Order of s0s1s2s1 : 4
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}*320b, {8,20}*320a
4-fold quotients : {4,20}*160, {8,10}*160
5-fold quotients : {8,8}*128c
8-fold quotients : {2,20}*80, {4,10}*80
10-fold quotients : {8,4}*64a, {4,8}*64b
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, {8,80}*1280a, {8,80}*1280b, {16,40}*1280c, {16,40}*1280e
3-fold covers : {8,120}*1920a, {24,40}*1920b
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)
( 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, 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)( 21,107)( 22,106)( 23,110)( 24,109)
( 25,108)( 26,102)( 27,101)( 28,105)( 29,104)( 30,103)( 31,117)( 32,116)
( 33,120)( 34,119)( 35,118)( 36,112)( 37,111)( 38,115)( 39,114)( 40,113)
( 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)( 61,142)( 62,141)( 63,145)( 64,144)
( 65,143)( 66,147)( 67,146)( 68,150)( 69,149)( 70,148)( 71,152)( 72,151)
( 73,155)( 74,154)( 75,153)( 76,157)( 77,156)( 78,160)( 79,159)( 80,158)
(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)(181,267)(182,266)(183,270)(184,269)
(185,268)(186,262)(187,261)(188,265)(189,264)(190,263)(191,277)(192,276)
(193,280)(194,279)(195,278)(196,272)(197,271)(198,275)(199,274)(200,273)
(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)(221,302)(222,301)(223,305)(224,304)
(225,303)(226,307)(227,306)(228,310)(229,309)(230,308)(231,312)(232,311)
(233,315)(234,314)(235,313)(236,317)(237,316)(238,320)(239,319)(240,318);;
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,
s2*s0*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*s0*s1*s2*s1 ];;
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)( 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, 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)( 21,107)( 22,106)( 23,110)
( 24,109)( 25,108)( 26,102)( 27,101)( 28,105)( 29,104)( 30,103)( 31,117)
( 32,116)( 33,120)( 34,119)( 35,118)( 36,112)( 37,111)( 38,115)( 39,114)
( 40,113)( 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)( 61,142)( 62,141)( 63,145)
( 64,144)( 65,143)( 66,147)( 67,146)( 68,150)( 69,149)( 70,148)( 71,152)
( 72,151)( 73,155)( 74,154)( 75,153)( 76,157)( 77,156)( 78,160)( 79,159)
( 80,158)(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)(181,267)(182,266)(183,270)
(184,269)(185,268)(186,262)(187,261)(188,265)(189,264)(190,263)(191,277)
(192,276)(193,280)(194,279)(195,278)(196,272)(197,271)(198,275)(199,274)
(200,273)(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)(221,302)(222,301)(223,305)
(224,304)(225,303)(226,307)(227,306)(228,310)(229,309)(230,308)(231,312)
(232,311)(233,315)(234,314)(235,313)(236,317)(237,316)(238,320)(239,319)
(240,318);
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,
s2*s0*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*s0*s1*s2*s1 >;

```
References : None.
to this polytope