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# Polytope of Type {36,4}

Atlas Canonical Name : {36,4}*576b
if this polytope has a name.
Group : SmallGroup(576,4968)
Rank : 3
Schlafli Type : {36,4}
Number of vertices, edges, etc : 72, 144, 8
Order of s0s1s2 : 36
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{36,4,2} of size 1152
Vertex Figure Of :
{2,36,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {36,4}*288b, {36,4}*288c, {18,4}*288
3-fold quotients : {12,4}*192b
4-fold quotients : {36,2}*144, {9,4}*144, {18,4}*144b, {18,4}*144c
6-fold quotients : {12,4}*96b, {12,4}*96c, {6,4}*96
8-fold quotients : {9,4}*72, {18,2}*72
12-fold quotients : {12,2}*48, {3,4}*48, {6,4}*48b, {6,4}*48c
16-fold quotients : {9,2}*36
24-fold quotients : {3,4}*24, {6,2}*24
36-fold quotients : {4,2}*16
48-fold quotients : {3,2}*12
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,4}*1152d, {36,8}*1152e, {36,8}*1152f, {72,4}*1152c, {72,4}*1152d
3-fold covers : {108,4}*1728b, {36,12}*1728e, {36,12}*1728f
Permutation Representation (GAP) :
```s0 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 33)( 14, 35)( 15, 34)
( 16, 36)( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 25)( 22, 27)( 23, 26)
( 24, 28)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 69)( 50, 71)
( 51, 70)( 52, 72)( 53, 65)( 54, 67)( 55, 66)( 56, 68)( 57, 61)( 58, 63)
( 59, 62)( 60, 64)( 73,109)( 74,111)( 75,110)( 76,112)( 77,117)( 78,119)
( 79,118)( 80,120)( 81,113)( 82,115)( 83,114)( 84,116)( 85,141)( 86,143)
( 87,142)( 88,144)( 89,137)( 90,139)( 91,138)( 92,140)( 93,133)( 94,135)
( 95,134)( 96,136)( 97,129)( 98,131)( 99,130)(100,132)(101,125)(102,127)
(103,126)(104,128)(105,121)(106,123)(107,122)(108,124)(146,147)(149,153)
(150,155)(151,154)(152,156)(157,177)(158,179)(159,178)(160,180)(161,173)
(162,175)(163,174)(164,176)(165,169)(166,171)(167,170)(168,172)(182,183)
(185,189)(186,191)(187,190)(188,192)(193,213)(194,215)(195,214)(196,216)
(197,209)(198,211)(199,210)(200,212)(201,205)(202,207)(203,206)(204,208)
(217,253)(218,255)(219,254)(220,256)(221,261)(222,263)(223,262)(224,264)
(225,257)(226,259)(227,258)(228,260)(229,285)(230,287)(231,286)(232,288)
(233,281)(234,283)(235,282)(236,284)(237,277)(238,279)(239,278)(240,280)
(241,273)(242,275)(243,274)(244,276)(245,269)(246,271)(247,270)(248,272)
(249,265)(250,267)(251,266)(252,268);;
s1 := (  1, 85)(  2, 86)(  3, 88)(  4, 87)(  5, 93)(  6, 94)(  7, 96)(  8, 95)
(  9, 89)( 10, 90)( 11, 92)( 12, 91)( 13, 73)( 14, 74)( 15, 76)( 16, 75)
( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 77)( 22, 78)( 23, 80)( 24, 79)
( 25,105)( 26,106)( 27,108)( 28,107)( 29,101)( 30,102)( 31,104)( 32,103)
( 33, 97)( 34, 98)( 35,100)( 36, 99)( 37,121)( 38,122)( 39,124)( 40,123)
( 41,129)( 42,130)( 43,132)( 44,131)( 45,125)( 46,126)( 47,128)( 48,127)
( 49,109)( 50,110)( 51,112)( 52,111)( 53,117)( 54,118)( 55,120)( 56,119)
( 57,113)( 58,114)( 59,116)( 60,115)( 61,141)( 62,142)( 63,144)( 64,143)
( 65,137)( 66,138)( 67,140)( 68,139)( 69,133)( 70,134)( 71,136)( 72,135)
(145,229)(146,230)(147,232)(148,231)(149,237)(150,238)(151,240)(152,239)
(153,233)(154,234)(155,236)(156,235)(157,217)(158,218)(159,220)(160,219)
(161,225)(162,226)(163,228)(164,227)(165,221)(166,222)(167,224)(168,223)
(169,249)(170,250)(171,252)(172,251)(173,245)(174,246)(175,248)(176,247)
(177,241)(178,242)(179,244)(180,243)(181,265)(182,266)(183,268)(184,267)
(185,273)(186,274)(187,276)(188,275)(189,269)(190,270)(191,272)(192,271)
(193,253)(194,254)(195,256)(196,255)(197,261)(198,262)(199,264)(200,263)
(201,257)(202,258)(203,260)(204,259)(205,285)(206,286)(207,288)(208,287)
(209,281)(210,282)(211,284)(212,283)(213,277)(214,278)(215,280)(216,279);;
s2 := (  1,148)(  2,147)(  3,146)(  4,145)(  5,152)(  6,151)(  7,150)(  8,149)
(  9,156)( 10,155)( 11,154)( 12,153)( 13,160)( 14,159)( 15,158)( 16,157)
( 17,164)( 18,163)( 19,162)( 20,161)( 21,168)( 22,167)( 23,166)( 24,165)
( 25,172)( 26,171)( 27,170)( 28,169)( 29,176)( 30,175)( 31,174)( 32,173)
( 33,180)( 34,179)( 35,178)( 36,177)( 37,184)( 38,183)( 39,182)( 40,181)
( 41,188)( 42,187)( 43,186)( 44,185)( 45,192)( 46,191)( 47,190)( 48,189)
( 49,196)( 50,195)( 51,194)( 52,193)( 53,200)( 54,199)( 55,198)( 56,197)
( 57,204)( 58,203)( 59,202)( 60,201)( 61,208)( 62,207)( 63,206)( 64,205)
( 65,212)( 66,211)( 67,210)( 68,209)( 69,216)( 70,215)( 71,214)( 72,213)
( 73,220)( 74,219)( 75,218)( 76,217)( 77,224)( 78,223)( 79,222)( 80,221)
( 81,228)( 82,227)( 83,226)( 84,225)( 85,232)( 86,231)( 87,230)( 88,229)
( 89,236)( 90,235)( 91,234)( 92,233)( 93,240)( 94,239)( 95,238)( 96,237)
( 97,244)( 98,243)( 99,242)(100,241)(101,248)(102,247)(103,246)(104,245)
(105,252)(106,251)(107,250)(108,249)(109,256)(110,255)(111,254)(112,253)
(113,260)(114,259)(115,258)(116,257)(117,264)(118,263)(119,262)(120,261)
(121,268)(122,267)(123,266)(124,265)(125,272)(126,271)(127,270)(128,269)
(129,276)(130,275)(131,274)(132,273)(133,280)(134,279)(135,278)(136,277)
(137,284)(138,283)(139,282)(140,281)(141,288)(142,287)(143,286)(144,285);;
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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope