# Polytope of Type {10,4}

Atlas Canonical Name : {10,4}*640b
if this polytope has a name.
Group : SmallGroup(640,21537)
Rank : 3
Schlafli Type : {10,4}
Number of vertices, edges, etc : 80, 160, 32
Order of s0s1s2 : 10
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 :
{10,4,2} of size 1280
Vertex Figure Of :
{2,10,4} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {5,4}*320, {10,4}*320a, {10,4}*320b
4-fold quotients : {5,4}*160
16-fold quotients : {10,2}*40
32-fold quotients : {5,2}*20
80-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {10,4}*1280a, {20,4}*1280b, {20,4}*1280c, {10,8}*1280c, {10,4}*1280b, {10,8}*1280d, {20,4}*1280d, {20,4}*1280e, {10,4}*1280c, {10,8}*1280e, {10,8}*1280f
3-fold covers : {10,12}*1920a, {30,4}*1920d
Permutation Representation (GAP) :
```s0 := (  2, 11)(  3, 14)(  4,  8)(  6, 15)(  7, 10)(  9, 13)( 17, 65)( 18, 75)
( 19, 78)( 20, 72)( 21, 69)( 22, 79)( 23, 74)( 24, 68)( 25, 77)( 26, 71)
( 27, 66)( 28, 76)( 29, 73)( 30, 67)( 31, 70)( 32, 80)( 33, 49)( 34, 59)
( 35, 62)( 36, 56)( 37, 53)( 38, 63)( 39, 58)( 40, 52)( 41, 61)( 42, 55)
( 43, 50)( 44, 60)( 45, 57)( 46, 51)( 47, 54)( 48, 64)( 82, 91)( 83, 94)
( 84, 88)( 86, 95)( 87, 90)( 89, 93)( 97,145)( 98,155)( 99,158)(100,152)
(101,149)(102,159)(103,154)(104,148)(105,157)(106,151)(107,146)(108,156)
(109,153)(110,147)(111,150)(112,160)(113,129)(114,139)(115,142)(116,136)
(117,133)(118,143)(119,138)(120,132)(121,141)(122,135)(123,130)(124,140)
(125,137)(126,131)(127,134)(128,144)(162,171)(163,174)(164,168)(166,175)
(167,170)(169,173)(177,225)(178,235)(179,238)(180,232)(181,229)(182,239)
(183,234)(184,228)(185,237)(186,231)(187,226)(188,236)(189,233)(190,227)
(191,230)(192,240)(193,209)(194,219)(195,222)(196,216)(197,213)(198,223)
(199,218)(200,212)(201,221)(202,215)(203,210)(204,220)(205,217)(206,211)
(207,214)(208,224)(242,251)(243,254)(244,248)(246,255)(247,250)(249,253)
(257,305)(258,315)(259,318)(260,312)(261,309)(262,319)(263,314)(264,308)
(265,317)(266,311)(267,306)(268,316)(269,313)(270,307)(271,310)(272,320)
(273,289)(274,299)(275,302)(276,296)(277,293)(278,303)(279,298)(280,292)
(281,301)(282,295)(283,290)(284,300)(285,297)(286,291)(287,294)(288,304);;
s1 := (  1,177)(  2,192)(  3,179)(  4,190)(  5,183)(  6,186)(  7,181)(  8,188)
(  9,187)( 10,182)( 11,185)( 12,184)( 13,189)( 14,180)( 15,191)( 16,178)
( 17,161)( 18,176)( 19,163)( 20,174)( 21,167)( 22,170)( 23,165)( 24,172)
( 25,171)( 26,166)( 27,169)( 28,168)( 29,173)( 30,164)( 31,175)( 32,162)
( 33,225)( 34,240)( 35,227)( 36,238)( 37,231)( 38,234)( 39,229)( 40,236)
( 41,235)( 42,230)( 43,233)( 44,232)( 45,237)( 46,228)( 47,239)( 48,226)
( 49,209)( 50,224)( 51,211)( 52,222)( 53,215)( 54,218)( 55,213)( 56,220)
( 57,219)( 58,214)( 59,217)( 60,216)( 61,221)( 62,212)( 63,223)( 64,210)
( 65,193)( 66,208)( 67,195)( 68,206)( 69,199)( 70,202)( 71,197)( 72,204)
( 73,203)( 74,198)( 75,201)( 76,200)( 77,205)( 78,196)( 79,207)( 80,194)
( 81,257)( 82,272)( 83,259)( 84,270)( 85,263)( 86,266)( 87,261)( 88,268)
( 89,267)( 90,262)( 91,265)( 92,264)( 93,269)( 94,260)( 95,271)( 96,258)
( 97,241)( 98,256)( 99,243)(100,254)(101,247)(102,250)(103,245)(104,252)
(105,251)(106,246)(107,249)(108,248)(109,253)(110,244)(111,255)(112,242)
(113,305)(114,320)(115,307)(116,318)(117,311)(118,314)(119,309)(120,316)
(121,315)(122,310)(123,313)(124,312)(125,317)(126,308)(127,319)(128,306)
(129,289)(130,304)(131,291)(132,302)(133,295)(134,298)(135,293)(136,300)
(137,299)(138,294)(139,297)(140,296)(141,301)(142,292)(143,303)(144,290)
(145,273)(146,288)(147,275)(148,286)(149,279)(150,282)(151,277)(152,284)
(153,283)(154,278)(155,281)(156,280)(157,285)(158,276)(159,287)(160,274);;
s2 := (  1, 85)(  2, 86)(  3, 87)(  4, 88)(  5, 81)(  6, 82)(  7, 83)(  8, 84)
(  9, 93)( 10, 94)( 11, 95)( 12, 96)( 13, 89)( 14, 90)( 15, 91)( 16, 92)
( 17,101)( 18,102)( 19,103)( 20,104)( 21, 97)( 22, 98)( 23, 99)( 24,100)
( 25,109)( 26,110)( 27,111)( 28,112)( 29,105)( 30,106)( 31,107)( 32,108)
( 33,117)( 34,118)( 35,119)( 36,120)( 37,113)( 38,114)( 39,115)( 40,116)
( 41,125)( 42,126)( 43,127)( 44,128)( 45,121)( 46,122)( 47,123)( 48,124)
( 49,133)( 50,134)( 51,135)( 52,136)( 53,129)( 54,130)( 55,131)( 56,132)
( 57,141)( 58,142)( 59,143)( 60,144)( 61,137)( 62,138)( 63,139)( 64,140)
( 65,149)( 66,150)( 67,151)( 68,152)( 69,145)( 70,146)( 71,147)( 72,148)
( 73,157)( 74,158)( 75,159)( 76,160)( 77,153)( 78,154)( 79,155)( 80,156)
(161,245)(162,246)(163,247)(164,248)(165,241)(166,242)(167,243)(168,244)
(169,253)(170,254)(171,255)(172,256)(173,249)(174,250)(175,251)(176,252)
(177,261)(178,262)(179,263)(180,264)(181,257)(182,258)(183,259)(184,260)
(185,269)(186,270)(187,271)(188,272)(189,265)(190,266)(191,267)(192,268)
(193,277)(194,278)(195,279)(196,280)(197,273)(198,274)(199,275)(200,276)
(201,285)(202,286)(203,287)(204,288)(205,281)(206,282)(207,283)(208,284)
(209,293)(210,294)(211,295)(212,296)(213,289)(214,290)(215,291)(216,292)
(217,301)(218,302)(219,303)(220,304)(221,297)(222,298)(223,299)(224,300)
(225,309)(226,310)(227,311)(228,312)(229,305)(230,306)(231,307)(232,308)
(233,317)(234,318)(235,319)(236,320)(237,313)(238,314)(239,315)(240,316);;
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*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*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*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope