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# Polytope of Type {5,5}

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

```
References : None.
to this polytope