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# Polytope of Type {8,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,18}*576b
if this polytope has a name.
Group : SmallGroup(576,4979)
Rank : 3
Schlafli Type : {8,18}
Number of vertices, edges, etc : 16, 144, 36
Order of s0s1s2 : 36
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{8,18,2} of size 1152
Vertex Figure Of :
{2,8,18} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,9}*288, {4,18}*288
3-fold quotients : {8,6}*192b
4-fold quotients : {4,9}*144, {4,18}*144b, {4,18}*144c
6-fold quotients : {8,3}*96, {4,6}*96
8-fold quotients : {4,9}*72, {2,18}*72
12-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
16-fold quotients : {2,9}*36
24-fold quotients : {4,3}*24, {2,6}*24
48-fold quotients : {2,3}*12
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,36}*1152e, {8,18}*1152f, {8,36}*1152h
3-fold covers : {8,54}*1728b, {24,18}*1728b, {24,18}*1728e
Permutation Representation (GAP) :
```s0 := (  1, 77)(  2, 78)(  3, 80)(  4, 79)(  5, 74)(  6, 73)(  7, 75)(  8, 76)
(  9, 85)( 10, 86)( 11, 88)( 12, 87)( 13, 82)( 14, 81)( 15, 83)( 16, 84)
( 17, 93)( 18, 94)( 19, 96)( 20, 95)( 21, 90)( 22, 89)( 23, 91)( 24, 92)
( 25,101)( 26,102)( 27,104)( 28,103)( 29, 98)( 30, 97)( 31, 99)( 32,100)
( 33,109)( 34,110)( 35,112)( 36,111)( 37,106)( 38,105)( 39,107)( 40,108)
( 41,117)( 42,118)( 43,120)( 44,119)( 45,114)( 46,113)( 47,115)( 48,116)
( 49,125)( 50,126)( 51,128)( 52,127)( 53,122)( 54,121)( 55,123)( 56,124)
( 57,133)( 58,134)( 59,136)( 60,135)( 61,130)( 62,129)( 63,131)( 64,132)
( 65,141)( 66,142)( 67,144)( 68,143)( 69,138)( 70,137)( 71,139)( 72,140)
(145,221)(146,222)(147,224)(148,223)(149,218)(150,217)(151,219)(152,220)
(153,229)(154,230)(155,232)(156,231)(157,226)(158,225)(159,227)(160,228)
(161,237)(162,238)(163,240)(164,239)(165,234)(166,233)(167,235)(168,236)
(169,245)(170,246)(171,248)(172,247)(173,242)(174,241)(175,243)(176,244)
(177,253)(178,254)(179,256)(180,255)(181,250)(182,249)(183,251)(184,252)
(185,261)(186,262)(187,264)(188,263)(189,258)(190,257)(191,259)(192,260)
(193,269)(194,270)(195,272)(196,271)(197,266)(198,265)(199,267)(200,268)
(201,277)(202,278)(203,280)(204,279)(205,274)(206,273)(207,275)(208,276)
(209,285)(210,286)(211,288)(212,287)(213,282)(214,281)(215,283)(216,284);;
s1 := (  3,  6)(  4,  5)(  7,  8)(  9, 17)( 10, 18)( 11, 22)( 12, 21)( 13, 20)
( 14, 19)( 15, 24)( 16, 23)( 25, 65)( 26, 66)( 27, 70)( 28, 69)( 29, 68)
( 30, 67)( 31, 72)( 32, 71)( 33, 57)( 34, 58)( 35, 62)( 36, 61)( 37, 60)
( 38, 59)( 39, 64)( 40, 63)( 41, 49)( 42, 50)( 43, 54)( 44, 53)( 45, 52)
( 46, 51)( 47, 56)( 48, 55)( 73, 74)( 75, 77)( 76, 78)( 81, 90)( 82, 89)
( 83, 93)( 84, 94)( 85, 91)( 86, 92)( 87, 95)( 88, 96)( 97,138)( 98,137)
( 99,141)(100,142)(101,139)(102,140)(103,143)(104,144)(105,130)(106,129)
(107,133)(108,134)(109,131)(110,132)(111,135)(112,136)(113,122)(114,121)
(115,125)(116,126)(117,123)(118,124)(119,127)(120,128)(147,150)(148,149)
(151,152)(153,161)(154,162)(155,166)(156,165)(157,164)(158,163)(159,168)
(160,167)(169,209)(170,210)(171,214)(172,213)(173,212)(174,211)(175,216)
(176,215)(177,201)(178,202)(179,206)(180,205)(181,204)(182,203)(183,208)
(184,207)(185,193)(186,194)(187,198)(188,197)(189,196)(190,195)(191,200)
(192,199)(217,218)(219,221)(220,222)(225,234)(226,233)(227,237)(228,238)
(229,235)(230,236)(231,239)(232,240)(241,282)(242,281)(243,285)(244,286)
(245,283)(246,284)(247,287)(248,288)(249,274)(250,273)(251,277)(252,278)
(253,275)(254,276)(255,279)(256,280)(257,266)(258,265)(259,269)(260,270)
(261,267)(262,268)(263,271)(264,272);;
s2 := (  1,193)(  2,194)(  3,199)(  4,200)(  5,198)(  6,197)(  7,195)(  8,196)
(  9,209)( 10,210)( 11,215)( 12,216)( 13,214)( 14,213)( 15,211)( 16,212)
( 17,201)( 18,202)( 19,207)( 20,208)( 21,206)( 22,205)( 23,203)( 24,204)
( 25,169)( 26,170)( 27,175)( 28,176)( 29,174)( 30,173)( 31,171)( 32,172)
( 33,185)( 34,186)( 35,191)( 36,192)( 37,190)( 38,189)( 39,187)( 40,188)
( 41,177)( 42,178)( 43,183)( 44,184)( 45,182)( 46,181)( 47,179)( 48,180)
( 49,145)( 50,146)( 51,151)( 52,152)( 53,150)( 54,149)( 55,147)( 56,148)
( 57,161)( 58,162)( 59,167)( 60,168)( 61,166)( 62,165)( 63,163)( 64,164)
( 65,153)( 66,154)( 67,159)( 68,160)( 69,158)( 70,157)( 71,155)( 72,156)
( 73,266)( 74,265)( 75,272)( 76,271)( 77,269)( 78,270)( 79,268)( 80,267)
( 81,282)( 82,281)( 83,288)( 84,287)( 85,285)( 86,286)( 87,284)( 88,283)
( 89,274)( 90,273)( 91,280)( 92,279)( 93,277)( 94,278)( 95,276)( 96,275)
( 97,242)( 98,241)( 99,248)(100,247)(101,245)(102,246)(103,244)(104,243)
(105,258)(106,257)(107,264)(108,263)(109,261)(110,262)(111,260)(112,259)
(113,250)(114,249)(115,256)(116,255)(117,253)(118,254)(119,252)(120,251)
(121,218)(122,217)(123,224)(124,223)(125,221)(126,222)(127,220)(128,219)
(129,234)(130,233)(131,240)(132,239)(133,237)(134,238)(135,236)(136,235)
(137,226)(138,225)(139,232)(140,231)(141,229)(142,230)(143,228)(144,227);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(288)!(  1, 77)(  2, 78)(  3, 80)(  4, 79)(  5, 74)(  6, 73)(  7, 75)
(  8, 76)(  9, 85)( 10, 86)( 11, 88)( 12, 87)( 13, 82)( 14, 81)( 15, 83)
( 16, 84)( 17, 93)( 18, 94)( 19, 96)( 20, 95)( 21, 90)( 22, 89)( 23, 91)
( 24, 92)( 25,101)( 26,102)( 27,104)( 28,103)( 29, 98)( 30, 97)( 31, 99)
( 32,100)( 33,109)( 34,110)( 35,112)( 36,111)( 37,106)( 38,105)( 39,107)
( 40,108)( 41,117)( 42,118)( 43,120)( 44,119)( 45,114)( 46,113)( 47,115)
( 48,116)( 49,125)( 50,126)( 51,128)( 52,127)( 53,122)( 54,121)( 55,123)
( 56,124)( 57,133)( 58,134)( 59,136)( 60,135)( 61,130)( 62,129)( 63,131)
( 64,132)( 65,141)( 66,142)( 67,144)( 68,143)( 69,138)( 70,137)( 71,139)
( 72,140)(145,221)(146,222)(147,224)(148,223)(149,218)(150,217)(151,219)
(152,220)(153,229)(154,230)(155,232)(156,231)(157,226)(158,225)(159,227)
(160,228)(161,237)(162,238)(163,240)(164,239)(165,234)(166,233)(167,235)
(168,236)(169,245)(170,246)(171,248)(172,247)(173,242)(174,241)(175,243)
(176,244)(177,253)(178,254)(179,256)(180,255)(181,250)(182,249)(183,251)
(184,252)(185,261)(186,262)(187,264)(188,263)(189,258)(190,257)(191,259)
(192,260)(193,269)(194,270)(195,272)(196,271)(197,266)(198,265)(199,267)
(200,268)(201,277)(202,278)(203,280)(204,279)(205,274)(206,273)(207,275)
(208,276)(209,285)(210,286)(211,288)(212,287)(213,282)(214,281)(215,283)
(216,284);
s1 := Sym(288)!(  3,  6)(  4,  5)(  7,  8)(  9, 17)( 10, 18)( 11, 22)( 12, 21)
( 13, 20)( 14, 19)( 15, 24)( 16, 23)( 25, 65)( 26, 66)( 27, 70)( 28, 69)
( 29, 68)( 30, 67)( 31, 72)( 32, 71)( 33, 57)( 34, 58)( 35, 62)( 36, 61)
( 37, 60)( 38, 59)( 39, 64)( 40, 63)( 41, 49)( 42, 50)( 43, 54)( 44, 53)
( 45, 52)( 46, 51)( 47, 56)( 48, 55)( 73, 74)( 75, 77)( 76, 78)( 81, 90)
( 82, 89)( 83, 93)( 84, 94)( 85, 91)( 86, 92)( 87, 95)( 88, 96)( 97,138)
( 98,137)( 99,141)(100,142)(101,139)(102,140)(103,143)(104,144)(105,130)
(106,129)(107,133)(108,134)(109,131)(110,132)(111,135)(112,136)(113,122)
(114,121)(115,125)(116,126)(117,123)(118,124)(119,127)(120,128)(147,150)
(148,149)(151,152)(153,161)(154,162)(155,166)(156,165)(157,164)(158,163)
(159,168)(160,167)(169,209)(170,210)(171,214)(172,213)(173,212)(174,211)
(175,216)(176,215)(177,201)(178,202)(179,206)(180,205)(181,204)(182,203)
(183,208)(184,207)(185,193)(186,194)(187,198)(188,197)(189,196)(190,195)
(191,200)(192,199)(217,218)(219,221)(220,222)(225,234)(226,233)(227,237)
(228,238)(229,235)(230,236)(231,239)(232,240)(241,282)(242,281)(243,285)
(244,286)(245,283)(246,284)(247,287)(248,288)(249,274)(250,273)(251,277)
(252,278)(253,275)(254,276)(255,279)(256,280)(257,266)(258,265)(259,269)
(260,270)(261,267)(262,268)(263,271)(264,272);
s2 := Sym(288)!(  1,193)(  2,194)(  3,199)(  4,200)(  5,198)(  6,197)(  7,195)
(  8,196)(  9,209)( 10,210)( 11,215)( 12,216)( 13,214)( 14,213)( 15,211)
( 16,212)( 17,201)( 18,202)( 19,207)( 20,208)( 21,206)( 22,205)( 23,203)
( 24,204)( 25,169)( 26,170)( 27,175)( 28,176)( 29,174)( 30,173)( 31,171)
( 32,172)( 33,185)( 34,186)( 35,191)( 36,192)( 37,190)( 38,189)( 39,187)
( 40,188)( 41,177)( 42,178)( 43,183)( 44,184)( 45,182)( 46,181)( 47,179)
( 48,180)( 49,145)( 50,146)( 51,151)( 52,152)( 53,150)( 54,149)( 55,147)
( 56,148)( 57,161)( 58,162)( 59,167)( 60,168)( 61,166)( 62,165)( 63,163)
( 64,164)( 65,153)( 66,154)( 67,159)( 68,160)( 69,158)( 70,157)( 71,155)
( 72,156)( 73,266)( 74,265)( 75,272)( 76,271)( 77,269)( 78,270)( 79,268)
( 80,267)( 81,282)( 82,281)( 83,288)( 84,287)( 85,285)( 86,286)( 87,284)
( 88,283)( 89,274)( 90,273)( 91,280)( 92,279)( 93,277)( 94,278)( 95,276)
( 96,275)( 97,242)( 98,241)( 99,248)(100,247)(101,245)(102,246)(103,244)
(104,243)(105,258)(106,257)(107,264)(108,263)(109,261)(110,262)(111,260)
(112,259)(113,250)(114,249)(115,256)(116,255)(117,253)(118,254)(119,252)
(120,251)(121,218)(122,217)(123,224)(124,223)(125,221)(126,222)(127,220)
(128,219)(129,234)(130,233)(131,240)(132,239)(133,237)(134,238)(135,236)
(136,235)(137,226)(138,225)(139,232)(140,231)(141,229)(142,230)(143,228)
(144,227);
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, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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