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# Polytope of Type {18,8}

Atlas Canonical Name : {18,8}*1152g
if this polytope has a name.
Group : SmallGroup(1152,154366)
Rank : 3
Schlafli Type : {18,8}
Number of vertices, edges, etc : 72, 288, 32
Order of s0s1s2 : 72
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,4}*576b
3-fold quotients : {6,8}*384g
4-fold quotients : {18,8}*288, {18,4}*288
6-fold quotients : {6,4}*192b
8-fold quotients : {18,4}*144a, {9,4}*144, {18,4}*144b, {18,4}*144c
12-fold quotients : {6,8}*96, {6,4}*96
16-fold quotients : {9,4}*72, {18,2}*72
24-fold quotients : {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
32-fold quotients : {9,2}*36
36-fold quotients : {2,8}*32
48-fold quotients : {3,4}*24, {6,2}*24
72-fold quotients : {2,4}*16
96-fold quotients : {3,2}*12
144-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 33)( 14, 34)( 15, 36)
( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)
( 24, 27)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 69)( 50, 70)
( 51, 72)( 52, 71)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 61)( 58, 62)
( 59, 64)( 60, 63)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 85,105)
( 86,106)( 87,108)( 88,107)( 89,101)( 90,102)( 91,104)( 92,103)( 93, 97)
( 94, 98)( 95,100)( 96, 99)(111,112)(113,117)(114,118)(115,120)(116,119)
(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)(127,140)(128,139)
(129,133)(130,134)(131,136)(132,135)(147,148)(149,153)(150,154)(151,156)
(152,155)(157,177)(158,178)(159,180)(160,179)(161,173)(162,174)(163,176)
(164,175)(165,169)(166,170)(167,172)(168,171)(183,184)(185,189)(186,190)
(187,192)(188,191)(193,213)(194,214)(195,216)(196,215)(197,209)(198,210)
(199,212)(200,211)(201,205)(202,206)(203,208)(204,207)(219,220)(221,225)
(222,226)(223,228)(224,227)(229,249)(230,250)(231,252)(232,251)(233,245)
(234,246)(235,248)(236,247)(237,241)(238,242)(239,244)(240,243)(255,256)
(257,261)(258,262)(259,264)(260,263)(265,285)(266,286)(267,288)(268,287)
(269,281)(270,282)(271,284)(272,283)(273,277)(274,278)(275,280)(276,279);;
s1 := (  1, 13)(  2, 16)(  3, 15)(  4, 14)(  5, 21)(  6, 24)(  7, 23)(  8, 22)
(  9, 17)( 10, 20)( 11, 19)( 12, 18)( 25, 33)( 26, 36)( 27, 35)( 28, 34)
( 30, 32)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)( 43, 59)
( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 61, 69)( 62, 72)( 63, 71)
( 64, 70)( 66, 68)( 73,121)( 74,124)( 75,123)( 76,122)( 77,129)( 78,132)
( 79,131)( 80,130)( 81,125)( 82,128)( 83,127)( 84,126)( 85,109)( 86,112)
( 87,111)( 88,110)( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)( 94,116)
( 95,115)( 96,114)( 97,141)( 98,144)( 99,143)(100,142)(101,137)(102,140)
(103,139)(104,138)(105,133)(106,136)(107,135)(108,134)(145,229)(146,232)
(147,231)(148,230)(149,237)(150,240)(151,239)(152,238)(153,233)(154,236)
(155,235)(156,234)(157,217)(158,220)(159,219)(160,218)(161,225)(162,228)
(163,227)(164,226)(165,221)(166,224)(167,223)(168,222)(169,249)(170,252)
(171,251)(172,250)(173,245)(174,248)(175,247)(176,246)(177,241)(178,244)
(179,243)(180,242)(181,265)(182,268)(183,267)(184,266)(185,273)(186,276)
(187,275)(188,274)(189,269)(190,272)(191,271)(192,270)(193,253)(194,256)
(195,255)(196,254)(197,261)(198,264)(199,263)(200,262)(201,257)(202,260)
(203,259)(204,258)(205,285)(206,288)(207,287)(208,286)(209,281)(210,284)
(211,283)(212,282)(213,277)(214,280)(215,279)(216,278);;
s2 := (  1,146)(  2,145)(  3,148)(  4,147)(  5,150)(  6,149)(  7,152)(  8,151)
(  9,154)( 10,153)( 11,156)( 12,155)( 13,158)( 14,157)( 15,160)( 16,159)
( 17,162)( 18,161)( 19,164)( 20,163)( 21,166)( 22,165)( 23,168)( 24,167)
( 25,170)( 26,169)( 27,172)( 28,171)( 29,174)( 30,173)( 31,176)( 32,175)
( 33,178)( 34,177)( 35,180)( 36,179)( 37,182)( 38,181)( 39,184)( 40,183)
( 41,186)( 42,185)( 43,188)( 44,187)( 45,190)( 46,189)( 47,192)( 48,191)
( 49,194)( 50,193)( 51,196)( 52,195)( 53,198)( 54,197)( 55,200)( 56,199)
( 57,202)( 58,201)( 59,204)( 60,203)( 61,206)( 62,205)( 63,208)( 64,207)
( 65,210)( 66,209)( 67,212)( 68,211)( 69,214)( 70,213)( 71,216)( 72,215)
( 73,254)( 74,253)( 75,256)( 76,255)( 77,258)( 78,257)( 79,260)( 80,259)
( 81,262)( 82,261)( 83,264)( 84,263)( 85,266)( 86,265)( 87,268)( 88,267)
( 89,270)( 90,269)( 91,272)( 92,271)( 93,274)( 94,273)( 95,276)( 96,275)
( 97,278)( 98,277)( 99,280)(100,279)(101,282)(102,281)(103,284)(104,283)
(105,286)(106,285)(107,288)(108,287)(109,218)(110,217)(111,220)(112,219)
(113,222)(114,221)(115,224)(116,223)(117,226)(118,225)(119,228)(120,227)
(121,230)(122,229)(123,232)(124,231)(125,234)(126,233)(127,236)(128,235)
(129,238)(130,237)(131,240)(132,239)(133,242)(134,241)(135,244)(136,243)
(137,246)(138,245)(139,248)(140,247)(141,250)(142,249)(143,252)(144,251);;
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*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope