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# Polytope of Type {4,10}

Atlas Canonical Name : {4,10}*640b
if this polytope has a name.
Group : SmallGroup(640,21537)
Rank : 3
Schlafli Type : {4,10}
Number of vertices, edges, etc : 32, 160, 80
Order of s0s1s2 : 10
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{4,10,2} of size 1280
Vertex Figure Of :
{2,4,10} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,5}*320, {4,10}*320a, {4,10}*320b
4-fold quotients : {4,5}*160
16-fold quotients : {2,10}*40
32-fold quotients : {2,5}*20
80-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,10}*1280a, {4,20}*1280b, {4,20}*1280c, {8,10}*1280c, {4,10}*1280b, {4,20}*1280d, {8,10}*1280d, {4,20}*1280e, {4,10}*1280c, {8,10}*1280e, {8,10}*1280f
3-fold covers : {12,10}*1920a, {4,30}*1920d
Permutation Representation (GAP) :
```s0 := (  1,169)(  2,170)(  3,171)(  4,172)(  5,173)(  6,174)(  7,175)(  8,176)
(  9,161)( 10,162)( 11,163)( 12,164)( 13,165)( 14,166)( 15,167)( 16,168)
( 17,185)( 18,186)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)( 24,192)
( 25,177)( 26,178)( 27,179)( 28,180)( 29,181)( 30,182)( 31,183)( 32,184)
( 33,201)( 34,202)( 35,203)( 36,204)( 37,205)( 38,206)( 39,207)( 40,208)
( 41,193)( 42,194)( 43,195)( 44,196)( 45,197)( 46,198)( 47,199)( 48,200)
( 49,217)( 50,218)( 51,219)( 52,220)( 53,221)( 54,222)( 55,223)( 56,224)
( 57,209)( 58,210)( 59,211)( 60,212)( 61,213)( 62,214)( 63,215)( 64,216)
( 65,233)( 66,234)( 67,235)( 68,236)( 69,237)( 70,238)( 71,239)( 72,240)
( 73,225)( 74,226)( 75,227)( 76,228)( 77,229)( 78,230)( 79,231)( 80,232)
( 81,249)( 82,250)( 83,251)( 84,252)( 85,253)( 86,254)( 87,255)( 88,256)
( 89,241)( 90,242)( 91,243)( 92,244)( 93,245)( 94,246)( 95,247)( 96,248)
( 97,265)( 98,266)( 99,267)(100,268)(101,269)(102,270)(103,271)(104,272)
(105,257)(106,258)(107,259)(108,260)(109,261)(110,262)(111,263)(112,264)
(113,281)(114,282)(115,283)(116,284)(117,285)(118,286)(119,287)(120,288)
(121,273)(122,274)(123,275)(124,276)(125,277)(126,278)(127,279)(128,280)
(129,297)(130,298)(131,299)(132,300)(133,301)(134,302)(135,303)(136,304)
(137,289)(138,290)(139,291)(140,292)(141,293)(142,294)(143,295)(144,296)
(145,313)(146,314)(147,315)(148,316)(149,317)(150,318)(151,319)(152,320)
(153,305)(154,306)(155,307)(156,308)(157,309)(158,310)(159,311)(160,312);;
s1 := (  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);;
s2 := (  1,145)(  2,157)(  3,152)(  4,156)(  5,154)(  6,150)(  7,159)(  8,147)
(  9,153)( 10,149)( 11,160)( 12,148)( 13,146)( 14,158)( 15,151)( 16,155)
( 17,129)( 18,141)( 19,136)( 20,140)( 21,138)( 22,134)( 23,143)( 24,131)
( 25,137)( 26,133)( 27,144)( 28,132)( 29,130)( 30,142)( 31,135)( 32,139)
( 33,113)( 34,125)( 35,120)( 36,124)( 37,122)( 38,118)( 39,127)( 40,115)
( 41,121)( 42,117)( 43,128)( 44,116)( 45,114)( 46,126)( 47,119)( 48,123)
( 49, 97)( 50,109)( 51,104)( 52,108)( 53,106)( 54,102)( 55,111)( 56, 99)
( 57,105)( 58,101)( 59,112)( 60,100)( 61, 98)( 62,110)( 63,103)( 64,107)
( 65, 81)( 66, 93)( 67, 88)( 68, 92)( 69, 90)( 70, 86)( 71, 95)( 72, 83)
( 73, 89)( 74, 85)( 75, 96)( 76, 84)( 77, 82)( 78, 94)( 79, 87)( 80, 91)
(161,305)(162,317)(163,312)(164,316)(165,314)(166,310)(167,319)(168,307)
(169,313)(170,309)(171,320)(172,308)(173,306)(174,318)(175,311)(176,315)
(177,289)(178,301)(179,296)(180,300)(181,298)(182,294)(183,303)(184,291)
(185,297)(186,293)(187,304)(188,292)(189,290)(190,302)(191,295)(192,299)
(193,273)(194,285)(195,280)(196,284)(197,282)(198,278)(199,287)(200,275)
(201,281)(202,277)(203,288)(204,276)(205,274)(206,286)(207,279)(208,283)
(209,257)(210,269)(211,264)(212,268)(213,266)(214,262)(215,271)(216,259)
(217,265)(218,261)(219,272)(220,260)(221,258)(222,270)(223,263)(224,267)
(225,241)(226,253)(227,248)(228,252)(229,250)(230,246)(231,255)(232,243)
(233,249)(234,245)(235,256)(236,244)(237,242)(238,254)(239,247)(240,251);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*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*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope