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# Polytope of Type {4,10}

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