Questions?
See the FAQ
or other info.

# Polytope of Type {8,40}

Atlas Canonical Name : {8,40}*640c
if this polytope has a name.
Group : SmallGroup(640,1225)
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}*320b
4-fold quotients : {4,20}*160
5-fold quotients : {8,8}*128d
8-fold quotients : {2,20}*80, {4,10}*80
10-fold quotients : {4,8}*64b, {8,4}*64b
16-fold quotients : {2,10}*40
20-fold quotients : {4,4}*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
3-fold covers : {8,120}*1920d, {24,40}*1920d
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)( 21,116)( 22,117)( 23,118)( 24,119)
( 25,120)( 26,111)( 27,112)( 28,113)( 29,114)( 30,115)( 31,106)( 32,107)
( 33,108)( 34,109)( 35,110)( 36,101)( 37,102)( 38,103)( 39,104)( 40,105)
( 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)( 61,151)( 62,152)( 63,153)( 64,154)
( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,141)( 72,142)
( 73,143)( 74,144)( 75,145)( 76,146)( 77,147)( 78,148)( 79,149)( 80,150)
(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)(181,276)(182,277)(183,278)(184,279)
(185,280)(186,271)(187,272)(188,273)(189,274)(190,275)(191,266)(192,267)
(193,268)(194,269)(195,270)(196,261)(197,262)(198,263)(199,264)(200,265)
(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)(221,311)(222,312)(223,313)(224,314)
(225,315)(226,316)(227,317)(228,318)(229,319)(230,320)(231,301)(232,302)
(233,303)(234,304)(235,305)(236,306)(237,307)(238,308)(239,309)(240,310);;
s1 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 21, 36)( 22, 40)( 23, 39)( 24, 38)( 25, 37)( 26, 31)( 27, 35)( 28, 34)
( 29, 33)( 30, 32)( 41, 51)( 42, 55)( 43, 54)( 44, 53)( 45, 52)( 46, 56)
( 47, 60)( 48, 59)( 49, 58)( 50, 57)( 61, 66)( 62, 70)( 63, 69)( 64, 68)
( 65, 67)( 71, 76)( 72, 80)( 73, 79)( 74, 78)( 75, 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)(181,236)(182,240)(183,239)(184,238)(185,237)(186,231)
(187,235)(188,234)(189,233)(190,232)(191,226)(192,230)(193,229)(194,228)
(195,227)(196,221)(197,225)(198,224)(199,223)(200,222)(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 := (  1,162)(  2,161)(  3,165)(  4,164)(  5,163)(  6,167)(  7,166)(  8,170)
(  9,169)( 10,168)( 11,172)( 12,171)( 13,175)( 14,174)( 15,173)( 16,177)
( 17,176)( 18,180)( 19,179)( 20,178)( 21,187)( 22,186)( 23,190)( 24,189)
( 25,188)( 26,182)( 27,181)( 28,185)( 29,184)( 30,183)( 31,197)( 32,196)
( 33,200)( 34,199)( 35,198)( 36,192)( 37,191)( 38,195)( 39,194)( 40,193)
( 41,212)( 42,211)( 43,215)( 44,214)( 45,213)( 46,217)( 47,216)( 48,220)
( 49,219)( 50,218)( 51,202)( 52,201)( 53,205)( 54,204)( 55,203)( 56,207)
( 57,206)( 58,210)( 59,209)( 60,208)( 61,237)( 62,236)( 63,240)( 64,239)
( 65,238)( 66,232)( 67,231)( 68,235)( 69,234)( 70,233)( 71,227)( 72,226)
( 73,230)( 74,229)( 75,228)( 76,222)( 77,221)( 78,225)( 79,224)( 80,223)
( 81,242)( 82,241)( 83,245)( 84,244)( 85,243)( 86,247)( 87,246)( 88,250)
( 89,249)( 90,248)( 91,252)( 92,251)( 93,255)( 94,254)( 95,253)( 96,257)
( 97,256)( 98,260)( 99,259)(100,258)(101,267)(102,266)(103,270)(104,269)
(105,268)(106,262)(107,261)(108,265)(109,264)(110,263)(111,277)(112,276)
(113,280)(114,279)(115,278)(116,272)(117,271)(118,275)(119,274)(120,273)
(121,292)(122,291)(123,295)(124,294)(125,293)(126,297)(127,296)(128,300)
(129,299)(130,298)(131,282)(132,281)(133,285)(134,284)(135,283)(136,287)
(137,286)(138,290)(139,289)(140,288)(141,317)(142,316)(143,320)(144,319)
(145,318)(146,312)(147,311)(148,315)(149,314)(150,313)(151,307)(152,306)
(153,310)(154,309)(155,308)(156,302)(157,301)(158,305)(159,304)(160,303);;
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*s0*s2*s1*s0*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*s2*s1*s2*s0*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)( 21,116)( 22,117)( 23,118)
( 24,119)( 25,120)( 26,111)( 27,112)( 28,113)( 29,114)( 30,115)( 31,106)
( 32,107)( 33,108)( 34,109)( 35,110)( 36,101)( 37,102)( 38,103)( 39,104)
( 40,105)( 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)( 61,151)( 62,152)( 63,153)
( 64,154)( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,141)
( 72,142)( 73,143)( 74,144)( 75,145)( 76,146)( 77,147)( 78,148)( 79,149)
( 80,150)(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)(181,276)(182,277)(183,278)
(184,279)(185,280)(186,271)(187,272)(188,273)(189,274)(190,275)(191,266)
(192,267)(193,268)(194,269)(195,270)(196,261)(197,262)(198,263)(199,264)
(200,265)(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)(221,311)(222,312)(223,313)
(224,314)(225,315)(226,316)(227,317)(228,318)(229,319)(230,320)(231,301)
(232,302)(233,303)(234,304)(235,305)(236,306)(237,307)(238,308)(239,309)
(240,310);
s1 := Sym(320)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 21, 36)( 22, 40)( 23, 39)( 24, 38)( 25, 37)( 26, 31)( 27, 35)
( 28, 34)( 29, 33)( 30, 32)( 41, 51)( 42, 55)( 43, 54)( 44, 53)( 45, 52)
( 46, 56)( 47, 60)( 48, 59)( 49, 58)( 50, 57)( 61, 66)( 62, 70)( 63, 69)
( 64, 68)( 65, 67)( 71, 76)( 72, 80)( 73, 79)( 74, 78)( 75, 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)(181,236)(182,240)(183,239)(184,238)(185,237)
(186,231)(187,235)(188,234)(189,233)(190,232)(191,226)(192,230)(193,229)
(194,228)(195,227)(196,221)(197,225)(198,224)(199,223)(200,222)(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)!(  1,162)(  2,161)(  3,165)(  4,164)(  5,163)(  6,167)(  7,166)
(  8,170)(  9,169)( 10,168)( 11,172)( 12,171)( 13,175)( 14,174)( 15,173)
( 16,177)( 17,176)( 18,180)( 19,179)( 20,178)( 21,187)( 22,186)( 23,190)
( 24,189)( 25,188)( 26,182)( 27,181)( 28,185)( 29,184)( 30,183)( 31,197)
( 32,196)( 33,200)( 34,199)( 35,198)( 36,192)( 37,191)( 38,195)( 39,194)
( 40,193)( 41,212)( 42,211)( 43,215)( 44,214)( 45,213)( 46,217)( 47,216)
( 48,220)( 49,219)( 50,218)( 51,202)( 52,201)( 53,205)( 54,204)( 55,203)
( 56,207)( 57,206)( 58,210)( 59,209)( 60,208)( 61,237)( 62,236)( 63,240)
( 64,239)( 65,238)( 66,232)( 67,231)( 68,235)( 69,234)( 70,233)( 71,227)
( 72,226)( 73,230)( 74,229)( 75,228)( 76,222)( 77,221)( 78,225)( 79,224)
( 80,223)( 81,242)( 82,241)( 83,245)( 84,244)( 85,243)( 86,247)( 87,246)
( 88,250)( 89,249)( 90,248)( 91,252)( 92,251)( 93,255)( 94,254)( 95,253)
( 96,257)( 97,256)( 98,260)( 99,259)(100,258)(101,267)(102,266)(103,270)
(104,269)(105,268)(106,262)(107,261)(108,265)(109,264)(110,263)(111,277)
(112,276)(113,280)(114,279)(115,278)(116,272)(117,271)(118,275)(119,274)
(120,273)(121,292)(122,291)(123,295)(124,294)(125,293)(126,297)(127,296)
(128,300)(129,299)(130,298)(131,282)(132,281)(133,285)(134,284)(135,283)
(136,287)(137,286)(138,290)(139,289)(140,288)(141,317)(142,316)(143,320)
(144,319)(145,318)(146,312)(147,311)(148,315)(149,314)(150,313)(151,307)
(152,306)(153,310)(154,309)(155,308)(156,302)(157,301)(158,305)(159,304)
(160,303);
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*s0*s2*s1*s0*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*s2*s1*s2*s0*s1*s2*s1*s2 >;

```
References : None.
to this polytope