# Polytope of Type {10,8}

Atlas Canonical Name : {10,8}*640b
if this polytope has a name.
Group : SmallGroup(640,21461)
Rank : 3
Schlafli Type : {10,8}
Number of vertices, edges, etc : 40, 160, 32
Order of s0s1s2 : 10
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{10,8,2} of size 1280
Vertex Figure Of :
{2,10,8} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {5,8}*320a, {10,4}*320a
4-fold quotients : {5,4}*160
Covers (Minimal Covers in Boldface) :
2-fold covers : {20,8}*1280j, {20,8}*1280l, {10,8}*1280c
3-fold covers : {30,8}*1920i
Permutation Representation (GAP) :
```s0 := (  3, 28)(  4, 27)(  5, 15)(  6, 16)(  7, 21)(  8, 22)(  9, 18)( 10, 17)
( 11, 12)( 13, 32)( 14, 31)( 23, 29)( 24, 30)( 25, 26)( 33,129)( 34,130)
( 35,156)( 36,155)( 37,143)( 38,144)( 39,149)( 40,150)( 41,146)( 42,145)
( 43,140)( 44,139)( 45,160)( 46,159)( 47,133)( 48,134)( 49,138)( 50,137)
( 51,147)( 52,148)( 53,135)( 54,136)( 55,157)( 56,158)( 57,154)( 58,153)
( 59,132)( 60,131)( 61,151)( 62,152)( 63,142)( 64,141)( 65, 97)( 66, 98)
( 67,124)( 68,123)( 69,111)( 70,112)( 71,117)( 72,118)( 73,114)( 74,113)
( 75,108)( 76,107)( 77,128)( 78,127)( 79,101)( 80,102)( 81,106)( 82,105)
( 83,115)( 84,116)( 85,103)( 86,104)( 87,125)( 88,126)( 89,122)( 90,121)
( 91,100)( 92, 99)( 93,119)( 94,120)( 95,110)( 96,109)(163,188)(164,187)
(165,175)(166,176)(167,181)(168,182)(169,178)(170,177)(171,172)(173,192)
(174,191)(183,189)(184,190)(185,186)(193,289)(194,290)(195,316)(196,315)
(197,303)(198,304)(199,309)(200,310)(201,306)(202,305)(203,300)(204,299)
(205,320)(206,319)(207,293)(208,294)(209,298)(210,297)(211,307)(212,308)
(213,295)(214,296)(215,317)(216,318)(217,314)(218,313)(219,292)(220,291)
(221,311)(222,312)(223,302)(224,301)(225,257)(226,258)(227,284)(228,283)
(229,271)(230,272)(231,277)(232,278)(233,274)(234,273)(235,268)(236,267)
(237,288)(238,287)(239,261)(240,262)(241,266)(242,265)(243,275)(244,276)
(245,263)(246,264)(247,285)(248,286)(249,282)(250,281)(251,260)(252,259)
(253,279)(254,280)(255,270)(256,269);;
s1 := (  1,193)(  2,194)(  3,215)(  4,216)(  5,217)(  6,218)(  7,207)(  8,208)
(  9,202)( 10,201)( 11,223)( 12,224)( 13,210)( 14,209)( 15,199)( 16,200)
( 17,206)( 18,205)( 19,219)( 20,220)( 21,214)( 22,213)( 23,195)( 24,196)
( 25,197)( 26,198)( 27,211)( 28,212)( 29,221)( 30,222)( 31,203)( 32,204)
( 33,161)( 34,162)( 35,183)( 36,184)( 37,185)( 38,186)( 39,175)( 40,176)
( 41,170)( 42,169)( 43,191)( 44,192)( 45,178)( 46,177)( 47,167)( 48,168)
( 49,174)( 50,173)( 51,187)( 52,188)( 53,182)( 54,181)( 55,163)( 56,164)
( 57,165)( 58,166)( 59,179)( 60,180)( 61,189)( 62,190)( 63,171)( 64,172)
( 65,289)( 66,290)( 67,311)( 68,312)( 69,313)( 70,314)( 71,303)( 72,304)
( 73,298)( 74,297)( 75,319)( 76,320)( 77,306)( 78,305)( 79,295)( 80,296)
( 81,302)( 82,301)( 83,315)( 84,316)( 85,310)( 86,309)( 87,291)( 88,292)
( 89,293)( 90,294)( 91,307)( 92,308)( 93,317)( 94,318)( 95,299)( 96,300)
( 97,257)( 98,258)( 99,279)(100,280)(101,281)(102,282)(103,271)(104,272)
(105,266)(106,265)(107,287)(108,288)(109,274)(110,273)(111,263)(112,264)
(113,270)(114,269)(115,283)(116,284)(117,278)(118,277)(119,259)(120,260)
(121,261)(122,262)(123,275)(124,276)(125,285)(126,286)(127,267)(128,268)
(129,225)(130,226)(131,247)(132,248)(133,249)(134,250)(135,239)(136,240)
(137,234)(138,233)(139,255)(140,256)(141,242)(142,241)(143,231)(144,232)
(145,238)(146,237)(147,251)(148,252)(149,246)(150,245)(151,227)(152,228)
(153,229)(154,230)(155,243)(156,244)(157,253)(158,254)(159,235)(160,236);;
s2 := (  1, 20)(  2, 19)(  3, 17)(  4, 18)(  5, 23)(  6, 24)(  7, 22)(  8, 21)
(  9, 27)( 10, 28)( 11, 26)( 12, 25)( 13, 32)( 14, 31)( 15, 29)( 16, 30)
( 33, 52)( 34, 51)( 35, 49)( 36, 50)( 37, 55)( 38, 56)( 39, 54)( 40, 53)
( 41, 59)( 42, 60)( 43, 58)( 44, 57)( 45, 64)( 46, 63)( 47, 61)( 48, 62)
( 65, 84)( 66, 83)( 67, 81)( 68, 82)( 69, 87)( 70, 88)( 71, 86)( 72, 85)
( 73, 91)( 74, 92)( 75, 90)( 76, 89)( 77, 96)( 78, 95)( 79, 93)( 80, 94)
( 97,116)( 98,115)( 99,113)(100,114)(101,119)(102,120)(103,118)(104,117)
(105,123)(106,124)(107,122)(108,121)(109,128)(110,127)(111,125)(112,126)
(129,148)(130,147)(131,145)(132,146)(133,151)(134,152)(135,150)(136,149)
(137,155)(138,156)(139,154)(140,153)(141,160)(142,159)(143,157)(144,158)
(161,180)(162,179)(163,177)(164,178)(165,183)(166,184)(167,182)(168,181)
(169,187)(170,188)(171,186)(172,185)(173,192)(174,191)(175,189)(176,190)
(193,212)(194,211)(195,209)(196,210)(197,215)(198,216)(199,214)(200,213)
(201,219)(202,220)(203,218)(204,217)(205,224)(206,223)(207,221)(208,222)
(225,244)(226,243)(227,241)(228,242)(229,247)(230,248)(231,246)(232,245)
(233,251)(234,252)(235,250)(236,249)(237,256)(238,255)(239,253)(240,254)
(257,276)(258,275)(259,273)(260,274)(261,279)(262,280)(263,278)(264,277)
(265,283)(266,284)(267,282)(268,281)(269,288)(270,287)(271,285)(272,286)
(289,308)(290,307)(291,305)(292,306)(293,311)(294,312)(295,310)(296,309)
(297,315)(298,316)(299,314)(300,313)(301,320)(302,319)(303,317)(304,318);;
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, s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s2,
s1*s2*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0,
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)!(  3, 28)(  4, 27)(  5, 15)(  6, 16)(  7, 21)(  8, 22)(  9, 18)
( 10, 17)( 11, 12)( 13, 32)( 14, 31)( 23, 29)( 24, 30)( 25, 26)( 33,129)
( 34,130)( 35,156)( 36,155)( 37,143)( 38,144)( 39,149)( 40,150)( 41,146)
( 42,145)( 43,140)( 44,139)( 45,160)( 46,159)( 47,133)( 48,134)( 49,138)
( 50,137)( 51,147)( 52,148)( 53,135)( 54,136)( 55,157)( 56,158)( 57,154)
( 58,153)( 59,132)( 60,131)( 61,151)( 62,152)( 63,142)( 64,141)( 65, 97)
( 66, 98)( 67,124)( 68,123)( 69,111)( 70,112)( 71,117)( 72,118)( 73,114)
( 74,113)( 75,108)( 76,107)( 77,128)( 78,127)( 79,101)( 80,102)( 81,106)
( 82,105)( 83,115)( 84,116)( 85,103)( 86,104)( 87,125)( 88,126)( 89,122)
( 90,121)( 91,100)( 92, 99)( 93,119)( 94,120)( 95,110)( 96,109)(163,188)
(164,187)(165,175)(166,176)(167,181)(168,182)(169,178)(170,177)(171,172)
(173,192)(174,191)(183,189)(184,190)(185,186)(193,289)(194,290)(195,316)
(196,315)(197,303)(198,304)(199,309)(200,310)(201,306)(202,305)(203,300)
(204,299)(205,320)(206,319)(207,293)(208,294)(209,298)(210,297)(211,307)
(212,308)(213,295)(214,296)(215,317)(216,318)(217,314)(218,313)(219,292)
(220,291)(221,311)(222,312)(223,302)(224,301)(225,257)(226,258)(227,284)
(228,283)(229,271)(230,272)(231,277)(232,278)(233,274)(234,273)(235,268)
(236,267)(237,288)(238,287)(239,261)(240,262)(241,266)(242,265)(243,275)
(244,276)(245,263)(246,264)(247,285)(248,286)(249,282)(250,281)(251,260)
(252,259)(253,279)(254,280)(255,270)(256,269);
s1 := Sym(320)!(  1,193)(  2,194)(  3,215)(  4,216)(  5,217)(  6,218)(  7,207)
(  8,208)(  9,202)( 10,201)( 11,223)( 12,224)( 13,210)( 14,209)( 15,199)
( 16,200)( 17,206)( 18,205)( 19,219)( 20,220)( 21,214)( 22,213)( 23,195)
( 24,196)( 25,197)( 26,198)( 27,211)( 28,212)( 29,221)( 30,222)( 31,203)
( 32,204)( 33,161)( 34,162)( 35,183)( 36,184)( 37,185)( 38,186)( 39,175)
( 40,176)( 41,170)( 42,169)( 43,191)( 44,192)( 45,178)( 46,177)( 47,167)
( 48,168)( 49,174)( 50,173)( 51,187)( 52,188)( 53,182)( 54,181)( 55,163)
( 56,164)( 57,165)( 58,166)( 59,179)( 60,180)( 61,189)( 62,190)( 63,171)
( 64,172)( 65,289)( 66,290)( 67,311)( 68,312)( 69,313)( 70,314)( 71,303)
( 72,304)( 73,298)( 74,297)( 75,319)( 76,320)( 77,306)( 78,305)( 79,295)
( 80,296)( 81,302)( 82,301)( 83,315)( 84,316)( 85,310)( 86,309)( 87,291)
( 88,292)( 89,293)( 90,294)( 91,307)( 92,308)( 93,317)( 94,318)( 95,299)
( 96,300)( 97,257)( 98,258)( 99,279)(100,280)(101,281)(102,282)(103,271)
(104,272)(105,266)(106,265)(107,287)(108,288)(109,274)(110,273)(111,263)
(112,264)(113,270)(114,269)(115,283)(116,284)(117,278)(118,277)(119,259)
(120,260)(121,261)(122,262)(123,275)(124,276)(125,285)(126,286)(127,267)
(128,268)(129,225)(130,226)(131,247)(132,248)(133,249)(134,250)(135,239)
(136,240)(137,234)(138,233)(139,255)(140,256)(141,242)(142,241)(143,231)
(144,232)(145,238)(146,237)(147,251)(148,252)(149,246)(150,245)(151,227)
(152,228)(153,229)(154,230)(155,243)(156,244)(157,253)(158,254)(159,235)
(160,236);
s2 := Sym(320)!(  1, 20)(  2, 19)(  3, 17)(  4, 18)(  5, 23)(  6, 24)(  7, 22)
(  8, 21)(  9, 27)( 10, 28)( 11, 26)( 12, 25)( 13, 32)( 14, 31)( 15, 29)
( 16, 30)( 33, 52)( 34, 51)( 35, 49)( 36, 50)( 37, 55)( 38, 56)( 39, 54)
( 40, 53)( 41, 59)( 42, 60)( 43, 58)( 44, 57)( 45, 64)( 46, 63)( 47, 61)
( 48, 62)( 65, 84)( 66, 83)( 67, 81)( 68, 82)( 69, 87)( 70, 88)( 71, 86)
( 72, 85)( 73, 91)( 74, 92)( 75, 90)( 76, 89)( 77, 96)( 78, 95)( 79, 93)
( 80, 94)( 97,116)( 98,115)( 99,113)(100,114)(101,119)(102,120)(103,118)
(104,117)(105,123)(106,124)(107,122)(108,121)(109,128)(110,127)(111,125)
(112,126)(129,148)(130,147)(131,145)(132,146)(133,151)(134,152)(135,150)
(136,149)(137,155)(138,156)(139,154)(140,153)(141,160)(142,159)(143,157)
(144,158)(161,180)(162,179)(163,177)(164,178)(165,183)(166,184)(167,182)
(168,181)(169,187)(170,188)(171,186)(172,185)(173,192)(174,191)(175,189)
(176,190)(193,212)(194,211)(195,209)(196,210)(197,215)(198,216)(199,214)
(200,213)(201,219)(202,220)(203,218)(204,217)(205,224)(206,223)(207,221)
(208,222)(225,244)(226,243)(227,241)(228,242)(229,247)(230,248)(231,246)
(232,245)(233,251)(234,252)(235,250)(236,249)(237,256)(238,255)(239,253)
(240,254)(257,276)(258,275)(259,273)(260,274)(261,279)(262,280)(263,278)
(264,277)(265,283)(266,284)(267,282)(268,281)(269,288)(270,287)(271,285)
(272,286)(289,308)(290,307)(291,305)(292,306)(293,311)(294,312)(295,310)
(296,309)(297,315)(298,316)(299,314)(300,313)(301,320)(302,319)(303,317)
(304,318);
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, s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s2,
s1*s2*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0,
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