# Polytope of Type {40,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {40,8}*640d
if this polytope has a name.
Group : SmallGroup(640,1265)
Rank : 3
Schlafli Type : {40,8}
Number of vertices, edges, etc : 40, 160, 8
Order of s0s1s2 : 40
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
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}*320b, {20,8}*320a
4-fold quotients : {20,4}*160, {10,8}*160
5-fold quotients : {8,8}*128a
8-fold quotients : {20,2}*80, {10,4}*80
10-fold quotients : {4,8}*64a, {8,4}*64b
16-fold quotients : {10,2}*40
20-fold quotients : {4,4}*32, {2,8}*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}*1280a, {80,8}*1280b, {40,16}*1280c, {40,16}*1280e
3-fold covers : {120,8}*1920a, {40,24}*1920b
Permutation Representation (GAP) :
```s0 := (  1, 81)(  2, 85)(  3, 84)(  4, 83)(  5, 82)(  6, 86)(  7, 90)(  8, 89)
(  9, 88)( 10, 87)( 11, 91)( 12, 95)( 13, 94)( 14, 93)( 15, 92)( 16, 96)
( 17,100)( 18, 99)( 19, 98)( 20, 97)( 21,106)( 22,110)( 23,109)( 24,108)
( 25,107)( 26,101)( 27,105)( 28,104)( 29,103)( 30,102)( 31,116)( 32,120)
( 33,119)( 34,118)( 35,117)( 36,111)( 37,115)( 38,114)( 39,113)( 40,112)
( 41,126)( 42,130)( 43,129)( 44,128)( 45,127)( 46,121)( 47,125)( 48,124)
( 49,123)( 50,122)( 51,136)( 52,140)( 53,139)( 54,138)( 55,137)( 56,131)
( 57,135)( 58,134)( 59,133)( 60,132)( 61,141)( 62,145)( 63,144)( 64,143)
( 65,142)( 66,146)( 67,150)( 68,149)( 69,148)( 70,147)( 71,151)( 72,155)
( 73,154)( 74,153)( 75,152)( 76,156)( 77,160)( 78,159)( 79,158)( 80,157)
(161,241)(162,245)(163,244)(164,243)(165,242)(166,246)(167,250)(168,249)
(169,248)(170,247)(171,251)(172,255)(173,254)(174,253)(175,252)(176,256)
(177,260)(178,259)(179,258)(180,257)(181,266)(182,270)(183,269)(184,268)
(185,267)(186,261)(187,265)(188,264)(189,263)(190,262)(191,276)(192,280)
(193,279)(194,278)(195,277)(196,271)(197,275)(198,274)(199,273)(200,272)
(201,286)(202,290)(203,289)(204,288)(205,287)(206,281)(207,285)(208,284)
(209,283)(210,282)(211,296)(212,300)(213,299)(214,298)(215,297)(216,291)
(217,295)(218,294)(219,293)(220,292)(221,301)(222,305)(223,304)(224,303)
(225,302)(226,306)(227,310)(228,309)(229,308)(230,307)(231,311)(232,315)
(233,314)(234,313)(235,312)(236,316)(237,320)(238,319)(239,318)(240,317);;
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 := (  1,161)(  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);;
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*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope