# Polytope of Type {10,5}

Atlas Canonical Name : {10,5}*320a
if this polytope has a name.
Group : SmallGroup(320,1582)
Rank : 3
Schlafli Type : {10,5}
Number of vertices, edges, etc : 32, 80, 16
Order of s0s1s2 : 8
Order of s0s1s2s1 : 5
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{10,5,2} of size 640
Vertex Figure Of :
{2,10,5} of size 640
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {5,5}*160
Covers (Minimal Covers in Boldface) :
2-fold covers : {10,5}*640, {10,10}*640a, {10,10}*640b
4-fold covers : {10,20}*1280a, {10,20}*1280b, {20,5}*1280, {10,10}*1280a
6-fold covers : {10,15}*1920, {30,10}*1920a
Permutation Representation (GAP) :
```s0 := (  1,161)(  2,162)(  3,181)(  4,182)(  5,187)(  6,188)(  7,175)(  8,176)
(  9,170)( 10,169)( 11,189)( 12,190)( 13,180)( 14,179)( 15,167)( 16,168)
( 17,185)( 18,186)( 19,174)( 20,173)( 21,163)( 22,164)( 23,184)( 24,183)
( 25,177)( 26,178)( 27,165)( 28,166)( 29,171)( 30,172)( 31,191)( 32,192)
( 33,289)( 34,290)( 35,309)( 36,310)( 37,315)( 38,316)( 39,303)( 40,304)
( 41,298)( 42,297)( 43,317)( 44,318)( 45,308)( 46,307)( 47,295)( 48,296)
( 49,313)( 50,314)( 51,302)( 52,301)( 53,291)( 54,292)( 55,312)( 56,311)
( 57,305)( 58,306)( 59,293)( 60,294)( 61,299)( 62,300)( 63,319)( 64,320)
( 65,257)( 66,258)( 67,277)( 68,278)( 69,283)( 70,284)( 71,271)( 72,272)
( 73,266)( 74,265)( 75,285)( 76,286)( 77,276)( 78,275)( 79,263)( 80,264)
( 81,281)( 82,282)( 83,270)( 84,269)( 85,259)( 86,260)( 87,280)( 88,279)
( 89,273)( 90,274)( 91,261)( 92,262)( 93,267)( 94,268)( 95,287)( 96,288)
( 97,225)( 98,226)( 99,245)(100,246)(101,251)(102,252)(103,239)(104,240)
(105,234)(106,233)(107,253)(108,254)(109,244)(110,243)(111,231)(112,232)
(113,249)(114,250)(115,238)(116,237)(117,227)(118,228)(119,248)(120,247)
(121,241)(122,242)(123,229)(124,230)(125,235)(126,236)(127,255)(128,256)
(129,193)(130,194)(131,213)(132,214)(133,219)(134,220)(135,207)(136,208)
(137,202)(138,201)(139,221)(140,222)(141,212)(142,211)(143,199)(144,200)
(145,217)(146,218)(147,206)(148,205)(149,195)(150,196)(151,216)(152,215)
(153,209)(154,210)(155,197)(156,198)(157,203)(158,204)(159,223)(160,224);;
s1 := (  1,194)(  2,193)(  3,224)(  4,223)(  5,197)(  6,198)(  7,220)(  8,219)
(  9,205)( 10,206)( 11,212)( 12,211)( 13,201)( 14,202)( 15,215)( 16,216)
( 17,213)( 18,214)( 19,204)( 20,203)( 21,209)( 22,210)( 23,207)( 24,208)
( 25,218)( 26,217)( 27,200)( 28,199)( 29,221)( 30,222)( 31,196)( 32,195)
( 33,162)( 34,161)( 35,192)( 36,191)( 37,165)( 38,166)( 39,188)( 40,187)
( 41,173)( 42,174)( 43,180)( 44,179)( 45,169)( 46,170)( 47,183)( 48,184)
( 49,181)( 50,182)( 51,172)( 52,171)( 53,177)( 54,178)( 55,175)( 56,176)
( 57,186)( 58,185)( 59,168)( 60,167)( 61,189)( 62,190)( 63,164)( 64,163)
( 65,290)( 66,289)( 67,320)( 68,319)( 69,293)( 70,294)( 71,316)( 72,315)
( 73,301)( 74,302)( 75,308)( 76,307)( 77,297)( 78,298)( 79,311)( 80,312)
( 81,309)( 82,310)( 83,300)( 84,299)( 85,305)( 86,306)( 87,303)( 88,304)
( 89,314)( 90,313)( 91,296)( 92,295)( 93,317)( 94,318)( 95,292)( 96,291)
( 97,258)( 98,257)( 99,288)(100,287)(101,261)(102,262)(103,284)(104,283)
(105,269)(106,270)(107,276)(108,275)(109,265)(110,266)(111,279)(112,280)
(113,277)(114,278)(115,268)(116,267)(117,273)(118,274)(119,271)(120,272)
(121,282)(122,281)(123,264)(124,263)(125,285)(126,286)(127,260)(128,259)
(129,226)(130,225)(131,256)(132,255)(133,229)(134,230)(135,252)(136,251)
(137,237)(138,238)(139,244)(140,243)(141,233)(142,234)(143,247)(144,248)
(145,245)(146,246)(147,236)(148,235)(149,241)(150,242)(151,239)(152,240)
(153,250)(154,249)(155,232)(156,231)(157,253)(158,254)(159,228)(160,227);;
s2 := (  1,192)(  2,191)(  3,171)(  4,172)(  5,165)(  6,166)(  7,178)(  8,177)
(  9,183)( 10,184)( 11,163)( 12,164)( 13,174)( 14,173)( 15,186)( 16,185)
( 17,168)( 18,167)( 19,180)( 20,179)( 21,189)( 22,190)( 23,169)( 24,170)
( 25,176)( 26,175)( 27,187)( 28,188)( 29,181)( 30,182)( 31,162)( 32,161)
( 33,320)( 34,319)( 35,299)( 36,300)( 37,293)( 38,294)( 39,306)( 40,305)
( 41,311)( 42,312)( 43,291)( 44,292)( 45,302)( 46,301)( 47,314)( 48,313)
( 49,296)( 50,295)( 51,308)( 52,307)( 53,317)( 54,318)( 55,297)( 56,298)
( 57,304)( 58,303)( 59,315)( 60,316)( 61,309)( 62,310)( 63,290)( 64,289)
( 65,288)( 66,287)( 67,267)( 68,268)( 69,261)( 70,262)( 71,274)( 72,273)
( 73,279)( 74,280)( 75,259)( 76,260)( 77,270)( 78,269)( 79,282)( 80,281)
( 81,264)( 82,263)( 83,276)( 84,275)( 85,285)( 86,286)( 87,265)( 88,266)
( 89,272)( 90,271)( 91,283)( 92,284)( 93,277)( 94,278)( 95,258)( 96,257)
( 97,256)( 98,255)( 99,235)(100,236)(101,229)(102,230)(103,242)(104,241)
(105,247)(106,248)(107,227)(108,228)(109,238)(110,237)(111,250)(112,249)
(113,232)(114,231)(115,244)(116,243)(117,253)(118,254)(119,233)(120,234)
(121,240)(122,239)(123,251)(124,252)(125,245)(126,246)(127,226)(128,225)
(129,224)(130,223)(131,203)(132,204)(133,197)(134,198)(135,210)(136,209)
(137,215)(138,216)(139,195)(140,196)(141,206)(142,205)(143,218)(144,217)
(145,200)(146,199)(147,212)(148,211)(149,221)(150,222)(151,201)(152,202)
(153,208)(154,207)(155,219)(156,220)(157,213)(158,214)(159,194)(160,193);;
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*s1*s2,
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope