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# Polytope of Type {4,10}

Atlas Canonical Name : {4,10}*320b
if this polytope has a name.
Group : SmallGroup(320,1636)
Rank : 3
Schlafli Type : {4,10}
Number of vertices, edges, etc : 16, 80, 40
Order of s0s1s2 : 5
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{4,10,2} of size 640
{4,10,3} of size 1920
{4,10,3} of size 1920
Vertex Figure Of :
{2,4,10} of size 640
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,5}*160
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,10}*640c, {8,10}*640d, {4,20}*640d, {4,20}*640e, {4,10}*640b
3-fold covers : {12,10}*960e, {4,30}*960d
4-fold covers : {8,20}*1280e, {8,20}*1280f, {8,20}*1280g, {8,20}*1280h, {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
5-fold covers : {4,50}*1600b, {20,10}*1600
6-fold covers : {12,20}*1920d, {12,20}*1920e, {24,10}*1920a, {24,10}*1920b, {8,30}*1920j, {8,30}*1920k, {4,60}*1920h, {4,60}*1920i, {12,10}*1920a, {4,30}*1920d
Permutation Representation (GAP) :
```s0 := (  1, 82)(  2, 81)(  3, 84)(  4, 83)(  5, 86)(  6, 85)(  7, 88)(  8, 87)
(  9, 90)( 10, 89)( 11, 92)( 12, 91)( 13, 94)( 14, 93)( 15, 96)( 16, 95)
( 17, 98)( 18, 97)( 19,100)( 20, 99)( 21,102)( 22,101)( 23,104)( 24,103)
( 25,106)( 26,105)( 27,108)( 28,107)( 29,110)( 30,109)( 31,112)( 32,111)
( 33,114)( 34,113)( 35,116)( 36,115)( 37,118)( 38,117)( 39,120)( 40,119)
( 41,122)( 42,121)( 43,124)( 44,123)( 45,126)( 46,125)( 47,128)( 48,127)
( 49,130)( 50,129)( 51,132)( 52,131)( 53,134)( 54,133)( 55,136)( 56,135)
( 57,138)( 58,137)( 59,140)( 60,139)( 61,142)( 62,141)( 63,144)( 64,143)
( 65,146)( 66,145)( 67,148)( 68,147)( 69,150)( 70,149)( 71,152)( 72,151)
( 73,154)( 74,153)( 75,156)( 76,155)( 77,158)( 78,157)( 79,160)( 80,159)
(161,242)(162,241)(163,244)(164,243)(165,246)(166,245)(167,248)(168,247)
(169,250)(170,249)(171,252)(172,251)(173,254)(174,253)(175,256)(176,255)
(177,258)(178,257)(179,260)(180,259)(181,262)(182,261)(183,264)(184,263)
(185,266)(186,265)(187,268)(188,267)(189,270)(190,269)(191,272)(192,271)
(193,274)(194,273)(195,276)(196,275)(197,278)(198,277)(199,280)(200,279)
(201,282)(202,281)(203,284)(204,283)(205,286)(206,285)(207,288)(208,287)
(209,290)(210,289)(211,292)(212,291)(213,294)(214,293)(215,296)(216,295)
(217,298)(218,297)(219,300)(220,299)(221,302)(222,301)(223,304)(224,303)
(225,306)(226,305)(227,308)(228,307)(229,310)(230,309)(231,312)(232,311)
(233,314)(234,313)(235,316)(236,315)(237,318)(238,317)(239,320)(240,319);;
s1 := (  1,161)(  2,171)(  3,174)(  4,168)(  5,165)(  6,175)(  7,170)(  8,164)
(  9,173)( 10,167)( 11,162)( 12,172)( 13,169)( 14,163)( 15,166)( 16,176)
( 17,225)( 18,235)( 19,238)( 20,232)( 21,229)( 22,239)( 23,234)( 24,228)
( 25,237)( 26,231)( 27,226)( 28,236)( 29,233)( 30,227)( 31,230)( 32,240)
( 33,209)( 34,219)( 35,222)( 36,216)( 37,213)( 38,223)( 39,218)( 40,212)
( 41,221)( 42,215)( 43,210)( 44,220)( 45,217)( 46,211)( 47,214)( 48,224)
( 49,193)( 50,203)( 51,206)( 52,200)( 53,197)( 54,207)( 55,202)( 56,196)
( 57,205)( 58,199)( 59,194)( 60,204)( 61,201)( 62,195)( 63,198)( 64,208)
( 65,177)( 66,187)( 67,190)( 68,184)( 69,181)( 70,191)( 71,186)( 72,180)
( 73,189)( 74,183)( 75,178)( 76,188)( 77,185)( 78,179)( 79,182)( 80,192)
( 81,241)( 82,251)( 83,254)( 84,248)( 85,245)( 86,255)( 87,250)( 88,244)
( 89,253)( 90,247)( 91,242)( 92,252)( 93,249)( 94,243)( 95,246)( 96,256)
( 97,305)( 98,315)( 99,318)(100,312)(101,309)(102,319)(103,314)(104,308)
(105,317)(106,311)(107,306)(108,316)(109,313)(110,307)(111,310)(112,320)
(113,289)(114,299)(115,302)(116,296)(117,293)(118,303)(119,298)(120,292)
(121,301)(122,295)(123,290)(124,300)(125,297)(126,291)(127,294)(128,304)
(129,273)(130,283)(131,286)(132,280)(133,277)(134,287)(135,282)(136,276)
(137,285)(138,279)(139,274)(140,284)(141,281)(142,275)(143,278)(144,288)
(145,257)(146,267)(147,270)(148,264)(149,261)(150,271)(151,266)(152,260)
(153,269)(154,263)(155,258)(156,268)(157,265)(158,259)(159,262)(160,272);;
s2 := (  1,273)(  2,274)(  3,276)(  4,275)(  5,278)(  6,277)(  7,279)(  8,280)
(  9,288)( 10,287)( 11,285)( 12,286)( 13,283)( 14,284)( 15,282)( 16,281)
( 17,257)( 18,258)( 19,260)( 20,259)( 21,262)( 22,261)( 23,263)( 24,264)
( 25,272)( 26,271)( 27,269)( 28,270)( 29,267)( 30,268)( 31,266)( 32,265)
( 33,241)( 34,242)( 35,244)( 36,243)( 37,246)( 38,245)( 39,247)( 40,248)
( 41,256)( 42,255)( 43,253)( 44,254)( 45,251)( 46,252)( 47,250)( 48,249)
( 49,305)( 50,306)( 51,308)( 52,307)( 53,310)( 54,309)( 55,311)( 56,312)
( 57,320)( 58,319)( 59,317)( 60,318)( 61,315)( 62,316)( 63,314)( 64,313)
( 65,289)( 66,290)( 67,292)( 68,291)( 69,294)( 70,293)( 71,295)( 72,296)
( 73,304)( 74,303)( 75,301)( 76,302)( 77,299)( 78,300)( 79,298)( 80,297)
( 81,193)( 82,194)( 83,196)( 84,195)( 85,198)( 86,197)( 87,199)( 88,200)
( 89,208)( 90,207)( 91,205)( 92,206)( 93,203)( 94,204)( 95,202)( 96,201)
( 97,177)( 98,178)( 99,180)(100,179)(101,182)(102,181)(103,183)(104,184)
(105,192)(106,191)(107,189)(108,190)(109,187)(110,188)(111,186)(112,185)
(113,161)(114,162)(115,164)(116,163)(117,166)(118,165)(119,167)(120,168)
(121,176)(122,175)(123,173)(124,174)(125,171)(126,172)(127,170)(128,169)
(129,225)(130,226)(131,228)(132,227)(133,230)(134,229)(135,231)(136,232)
(137,240)(138,239)(139,237)(140,238)(141,235)(142,236)(143,234)(144,233)
(145,209)(146,210)(147,212)(148,211)(149,214)(150,213)(151,215)(152,216)
(153,224)(154,223)(155,221)(156,222)(157,219)(158,220)(159,218)(160,217);;
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope