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# Polytope of Type {5,4}

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

```
References : None.
to this polytope