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# Polytope of Type {4,40}

Atlas Canonical Name : {4,40}*640a
if this polytope has a name.
Group : SmallGroup(640,981)
Rank : 3
Schlafli Type : {4,40}
Number of vertices, edges, etc : 8, 160, 80
Order of s0s1s2 : 40
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,40,2} of size 1280
Vertex Figure Of :
{2,4,40} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,40}*320a, {4,20}*320, {4,40}*320b
4-fold quotients : {4,20}*160, {2,40}*160
5-fold quotients : {4,8}*128a
8-fold quotients : {2,20}*80, {4,10}*80
10-fold quotients : {4,8}*64a, {4,8}*64b, {4,4}*64
16-fold quotients : {2,10}*40
20-fold quotients : {4,4}*32, {2,8}*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, {4,40}*1280a, {8,40}*1280d, {4,80}*1280a, {4,80}*1280b
3-fold covers : {4,120}*1920a, {12,40}*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)( 21,101)( 22,102)( 23,103)( 24,104)
( 25,105)( 26,106)( 27,107)( 28,108)( 29,109)( 30,110)( 31,111)( 32,112)
( 33,113)( 34,114)( 35,115)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)
( 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)( 61,146)( 62,147)( 63,148)( 64,149)
( 65,150)( 66,141)( 67,142)( 68,143)( 69,144)( 70,145)( 71,156)( 72,157)
( 73,158)( 74,159)( 75,160)( 76,151)( 77,152)( 78,153)( 79,154)( 80,155)
(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)(181,261)(182,262)(183,263)(184,264)
(185,265)(186,266)(187,267)(188,268)(189,269)(190,270)(191,271)(192,272)
(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)(200,280)
(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)(221,306)(222,307)(223,308)(224,309)
(225,310)(226,301)(227,302)(228,303)(229,304)(230,305)(231,316)(232,317)
(233,318)(234,319)(235,320)(236,311)(237,312)(238,313)(239,314)(240,315);;
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 := (  1,162)(  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, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*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)( 21,101)( 22,102)( 23,103)
( 24,104)( 25,105)( 26,106)( 27,107)( 28,108)( 29,109)( 30,110)( 31,111)
( 32,112)( 33,113)( 34,114)( 35,115)( 36,116)( 37,117)( 38,118)( 39,119)
( 40,120)( 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)( 61,146)( 62,147)( 63,148)
( 64,149)( 65,150)( 66,141)( 67,142)( 68,143)( 69,144)( 70,145)( 71,156)
( 72,157)( 73,158)( 74,159)( 75,160)( 76,151)( 77,152)( 78,153)( 79,154)
( 80,155)(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)(181,261)(182,262)(183,263)
(184,264)(185,265)(186,266)(187,267)(188,268)(189,269)(190,270)(191,271)
(192,272)(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)
(200,280)(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)(221,306)(222,307)(223,308)
(224,309)(225,310)(226,301)(227,302)(228,303)(229,304)(230,305)(231,316)
(232,317)(233,318)(234,319)(235,320)(236,311)(237,312)(238,313)(239,314)
(240,315);
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)!(  1,162)(  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, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*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