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# Polytope of Type {40,4}

Atlas Canonical Name : {40,4}*640a
if this polytope has a name.
Group : SmallGroup(640,981)
Rank : 3
Schlafli Type : {40,4}
Number of vertices, edges, etc : 80, 160, 8
Order of s0s1s2 : 40
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{40,4,2} of size 1280
Vertex Figure Of :
{2,40,4} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {40,4}*320a, {20,4}*320, {40,4}*320b
4-fold quotients : {20,4}*160, {40,2}*160
5-fold quotients : {8,4}*128a
8-fold quotients : {20,2}*80, {10,4}*80
10-fold quotients : {8,4}*64a, {8,4}*64b, {4,4}*64
16-fold quotients : {10,2}*40
20-fold quotients : {4,4}*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, {40,4}*1280a, {40,8}*1280d, {80,4}*1280a, {80,4}*1280b
3-fold covers : {120,4}*1920a, {40,12}*1920a
Permutation Representation (GAP) :
```s0 := (  1,161)(  2,165)(  3,164)(  4,163)(  5,162)(  6,166)(  7,170)(  8,169)
(  9,168)( 10,167)( 11,171)( 12,175)( 13,174)( 14,173)( 15,172)( 16,176)
( 17,180)( 18,179)( 19,178)( 20,177)( 21,186)( 22,190)( 23,189)( 24,188)
( 25,187)( 26,181)( 27,185)( 28,184)( 29,183)( 30,182)( 31,196)( 32,200)
( 33,199)( 34,198)( 35,197)( 36,191)( 37,195)( 38,194)( 39,193)( 40,192)
( 41,211)( 42,215)( 43,214)( 44,213)( 45,212)( 46,216)( 47,220)( 48,219)
( 49,218)( 50,217)( 51,201)( 52,205)( 53,204)( 54,203)( 55,202)( 56,206)
( 57,210)( 58,209)( 59,208)( 60,207)( 61,236)( 62,240)( 63,239)( 64,238)
( 65,237)( 66,231)( 67,235)( 68,234)( 69,233)( 70,232)( 71,226)( 72,230)
( 73,229)( 74,228)( 75,227)( 76,221)( 77,225)( 78,224)( 79,223)( 80,222)
( 81,241)( 82,245)( 83,244)( 84,243)( 85,242)( 86,246)( 87,250)( 88,249)
( 89,248)( 90,247)( 91,251)( 92,255)( 93,254)( 94,253)( 95,252)( 96,256)
( 97,260)( 98,259)( 99,258)(100,257)(101,266)(102,270)(103,269)(104,268)
(105,267)(106,261)(107,265)(108,264)(109,263)(110,262)(111,276)(112,280)
(113,279)(114,278)(115,277)(116,271)(117,275)(118,274)(119,273)(120,272)
(121,291)(122,295)(123,294)(124,293)(125,292)(126,296)(127,300)(128,299)
(129,298)(130,297)(131,281)(132,285)(133,284)(134,283)(135,282)(136,286)
(137,290)(138,289)(139,288)(140,287)(141,316)(142,320)(143,319)(144,318)
(145,317)(146,311)(147,315)(148,314)(149,313)(150,312)(151,306)(152,310)
(153,309)(154,308)(155,307)(156,301)(157,305)(158,304)(159,303)(160,302);;
s1 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)( 18, 20)
( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)( 38, 40)
( 41, 52)( 42, 51)( 43, 55)( 44, 54)( 45, 53)( 46, 57)( 47, 56)( 48, 60)
( 49, 59)( 50, 58)( 61, 72)( 62, 71)( 63, 75)( 64, 74)( 65, 73)( 66, 77)
( 67, 76)( 68, 80)( 69, 79)( 70, 78)( 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)
(181,222)(182,221)(183,225)(184,224)(185,223)(186,227)(187,226)(188,230)
(189,229)(190,228)(191,232)(192,231)(193,235)(194,234)(195,233)(196,237)
(197,236)(198,240)(199,239)(200,238)(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)( 21,101)( 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);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*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*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope