# Polytope of Type {10,10}

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

```
References : None.
to this polytope