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# Polytope of Type {18,12}

Atlas Canonical Name : {18,12}*1296i
if this polytope has a name.
Group : SmallGroup(1296,1786)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 54, 324, 36
Order of s0s1s2 : 3
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,12}*432d
4-fold quotients : {18,6}*324c
9-fold quotients : {6,12}*144d
12-fold quotients : {6,6}*108
27-fold quotients : {6,4}*48b
54-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 25)( 14, 26)( 15, 28)
( 16, 27)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 29)( 22, 30)( 23, 32)
( 24, 31)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 61)( 50, 62)
( 51, 64)( 52, 63)( 53, 69)( 54, 70)( 55, 72)( 56, 71)( 57, 65)( 58, 66)
( 59, 68)( 60, 67)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 85, 97)
( 86, 98)( 87,100)( 88, 99)( 89,105)( 90,106)( 91,108)( 92,107)( 93,101)
( 94,102)( 95,104)( 96,103)(109,225)(110,226)(111,228)(112,227)(113,221)
(114,222)(115,224)(116,223)(117,217)(118,218)(119,220)(120,219)(121,249)
(122,250)(123,252)(124,251)(125,245)(126,246)(127,248)(128,247)(129,241)
(130,242)(131,244)(132,243)(133,237)(134,238)(135,240)(136,239)(137,233)
(138,234)(139,236)(140,235)(141,229)(142,230)(143,232)(144,231)(145,261)
(146,262)(147,264)(148,263)(149,257)(150,258)(151,260)(152,259)(153,253)
(154,254)(155,256)(156,255)(157,285)(158,286)(159,288)(160,287)(161,281)
(162,282)(163,284)(164,283)(165,277)(166,278)(167,280)(168,279)(169,273)
(170,274)(171,276)(172,275)(173,269)(174,270)(175,272)(176,271)(177,265)
(178,266)(179,268)(180,267)(181,297)(182,298)(183,300)(184,299)(185,293)
(186,294)(187,296)(188,295)(189,289)(190,290)(191,292)(192,291)(193,321)
(194,322)(195,324)(196,323)(197,317)(198,318)(199,320)(200,319)(201,313)
(202,314)(203,316)(204,315)(205,309)(206,310)(207,312)(208,311)(209,305)
(210,306)(211,308)(212,307)(213,301)(214,302)(215,304)(216,303);;
s1 := (  1,109)(  2,112)(  3,111)(  4,110)(  5,117)(  6,120)(  7,119)(  8,118)
(  9,113)( 10,116)( 11,115)( 12,114)( 13,129)( 14,132)( 15,131)( 16,130)
( 17,125)( 18,128)( 19,127)( 20,126)( 21,121)( 22,124)( 23,123)( 24,122)
( 25,137)( 26,140)( 27,139)( 28,138)( 29,133)( 30,136)( 31,135)( 32,134)
( 33,141)( 34,144)( 35,143)( 36,142)( 37,209)( 38,212)( 39,211)( 40,210)
( 41,205)( 42,208)( 43,207)( 44,206)( 45,213)( 46,216)( 47,215)( 48,214)
( 49,181)( 50,184)( 51,183)( 52,182)( 53,189)( 54,192)( 55,191)( 56,190)
( 57,185)( 58,188)( 59,187)( 60,186)( 61,201)( 62,204)( 63,203)( 64,202)
( 65,197)( 66,200)( 67,199)( 68,198)( 69,193)( 70,196)( 71,195)( 72,194)
( 73,157)( 74,160)( 75,159)( 76,158)( 77,165)( 78,168)( 79,167)( 80,166)
( 81,161)( 82,164)( 83,163)( 84,162)( 85,177)( 86,180)( 87,179)( 88,178)
( 89,173)( 90,176)( 91,175)( 92,174)( 93,169)( 94,172)( 95,171)( 96,170)
( 97,149)( 98,152)( 99,151)(100,150)(101,145)(102,148)(103,147)(104,146)
(105,153)(106,156)(107,155)(108,154)(217,225)(218,228)(219,227)(220,226)
(222,224)(229,233)(230,236)(231,235)(232,234)(238,240)(242,244)(245,249)
(246,252)(247,251)(248,250)(253,313)(254,316)(255,315)(256,314)(257,321)
(258,324)(259,323)(260,322)(261,317)(262,320)(263,319)(264,318)(265,297)
(266,300)(267,299)(268,298)(269,293)(270,296)(271,295)(272,294)(273,289)
(274,292)(275,291)(276,290)(277,305)(278,308)(279,307)(280,306)(281,301)
(282,304)(283,303)(284,302)(285,309)(286,312)(287,311)(288,310);;
s2 := (  1, 38)(  2, 37)(  3, 40)(  4, 39)(  5, 42)(  6, 41)(  7, 44)(  8, 43)
(  9, 46)( 10, 45)( 11, 48)( 12, 47)( 13, 62)( 14, 61)( 15, 64)( 16, 63)
( 17, 66)( 18, 65)( 19, 68)( 20, 67)( 21, 70)( 22, 69)( 23, 72)( 24, 71)
( 25, 50)( 26, 49)( 27, 52)( 28, 51)( 29, 54)( 30, 53)( 31, 56)( 32, 55)
( 33, 58)( 34, 57)( 35, 60)( 36, 59)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 98)( 86, 97)( 87,100)( 88, 99)( 89,102)( 90,101)
( 91,104)( 92,103)( 93,106)( 94,105)( 95,108)( 96,107)(109,146)(110,145)
(111,148)(112,147)(113,150)(114,149)(115,152)(116,151)(117,154)(118,153)
(119,156)(120,155)(121,170)(122,169)(123,172)(124,171)(125,174)(126,173)
(127,176)(128,175)(129,178)(130,177)(131,180)(132,179)(133,158)(134,157)
(135,160)(136,159)(137,162)(138,161)(139,164)(140,163)(141,166)(142,165)
(143,168)(144,167)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)
(193,206)(194,205)(195,208)(196,207)(197,210)(198,209)(199,212)(200,211)
(201,214)(202,213)(203,216)(204,215)(217,254)(218,253)(219,256)(220,255)
(221,258)(222,257)(223,260)(224,259)(225,262)(226,261)(227,264)(228,263)
(229,278)(230,277)(231,280)(232,279)(233,282)(234,281)(235,284)(236,283)
(237,286)(238,285)(239,288)(240,287)(241,266)(242,265)(243,268)(244,267)
(245,270)(246,269)(247,272)(248,271)(249,274)(250,273)(251,276)(252,275)
(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,314)(302,313)
(303,316)(304,315)(305,318)(306,317)(307,320)(308,319)(309,322)(310,321)
(311,324)(312,323);;
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*s0*s1*s2*s0*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(324)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 25)( 14, 26)
( 15, 28)( 16, 27)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 29)( 22, 30)
( 23, 32)( 24, 31)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 61)
( 50, 62)( 51, 64)( 52, 63)( 53, 69)( 54, 70)( 55, 72)( 56, 71)( 57, 65)
( 58, 66)( 59, 68)( 60, 67)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)
( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,105)( 90,106)( 91,108)( 92,107)
( 93,101)( 94,102)( 95,104)( 96,103)(109,225)(110,226)(111,228)(112,227)
(113,221)(114,222)(115,224)(116,223)(117,217)(118,218)(119,220)(120,219)
(121,249)(122,250)(123,252)(124,251)(125,245)(126,246)(127,248)(128,247)
(129,241)(130,242)(131,244)(132,243)(133,237)(134,238)(135,240)(136,239)
(137,233)(138,234)(139,236)(140,235)(141,229)(142,230)(143,232)(144,231)
(145,261)(146,262)(147,264)(148,263)(149,257)(150,258)(151,260)(152,259)
(153,253)(154,254)(155,256)(156,255)(157,285)(158,286)(159,288)(160,287)
(161,281)(162,282)(163,284)(164,283)(165,277)(166,278)(167,280)(168,279)
(169,273)(170,274)(171,276)(172,275)(173,269)(174,270)(175,272)(176,271)
(177,265)(178,266)(179,268)(180,267)(181,297)(182,298)(183,300)(184,299)
(185,293)(186,294)(187,296)(188,295)(189,289)(190,290)(191,292)(192,291)
(193,321)(194,322)(195,324)(196,323)(197,317)(198,318)(199,320)(200,319)
(201,313)(202,314)(203,316)(204,315)(205,309)(206,310)(207,312)(208,311)
(209,305)(210,306)(211,308)(212,307)(213,301)(214,302)(215,304)(216,303);
s1 := Sym(324)!(  1,109)(  2,112)(  3,111)(  4,110)(  5,117)(  6,120)(  7,119)
(  8,118)(  9,113)( 10,116)( 11,115)( 12,114)( 13,129)( 14,132)( 15,131)
( 16,130)( 17,125)( 18,128)( 19,127)( 20,126)( 21,121)( 22,124)( 23,123)
( 24,122)( 25,137)( 26,140)( 27,139)( 28,138)( 29,133)( 30,136)( 31,135)
( 32,134)( 33,141)( 34,144)( 35,143)( 36,142)( 37,209)( 38,212)( 39,211)
( 40,210)( 41,205)( 42,208)( 43,207)( 44,206)( 45,213)( 46,216)( 47,215)
( 48,214)( 49,181)( 50,184)( 51,183)( 52,182)( 53,189)( 54,192)( 55,191)
( 56,190)( 57,185)( 58,188)( 59,187)( 60,186)( 61,201)( 62,204)( 63,203)
( 64,202)( 65,197)( 66,200)( 67,199)( 68,198)( 69,193)( 70,196)( 71,195)
( 72,194)( 73,157)( 74,160)( 75,159)( 76,158)( 77,165)( 78,168)( 79,167)
( 80,166)( 81,161)( 82,164)( 83,163)( 84,162)( 85,177)( 86,180)( 87,179)
( 88,178)( 89,173)( 90,176)( 91,175)( 92,174)( 93,169)( 94,172)( 95,171)
( 96,170)( 97,149)( 98,152)( 99,151)(100,150)(101,145)(102,148)(103,147)
(104,146)(105,153)(106,156)(107,155)(108,154)(217,225)(218,228)(219,227)
(220,226)(222,224)(229,233)(230,236)(231,235)(232,234)(238,240)(242,244)
(245,249)(246,252)(247,251)(248,250)(253,313)(254,316)(255,315)(256,314)
(257,321)(258,324)(259,323)(260,322)(261,317)(262,320)(263,319)(264,318)
(265,297)(266,300)(267,299)(268,298)(269,293)(270,296)(271,295)(272,294)
(273,289)(274,292)(275,291)(276,290)(277,305)(278,308)(279,307)(280,306)
(281,301)(282,304)(283,303)(284,302)(285,309)(286,312)(287,311)(288,310);
s2 := Sym(324)!(  1, 38)(  2, 37)(  3, 40)(  4, 39)(  5, 42)(  6, 41)(  7, 44)
(  8, 43)(  9, 46)( 10, 45)( 11, 48)( 12, 47)( 13, 62)( 14, 61)( 15, 64)
( 16, 63)( 17, 66)( 18, 65)( 19, 68)( 20, 67)( 21, 70)( 22, 69)( 23, 72)
( 24, 71)( 25, 50)( 26, 49)( 27, 52)( 28, 51)( 29, 54)( 30, 53)( 31, 56)
( 32, 55)( 33, 58)( 34, 57)( 35, 60)( 36, 59)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 98)( 86, 97)( 87,100)( 88, 99)( 89,102)
( 90,101)( 91,104)( 92,103)( 93,106)( 94,105)( 95,108)( 96,107)(109,146)
(110,145)(111,148)(112,147)(113,150)(114,149)(115,152)(116,151)(117,154)
(118,153)(119,156)(120,155)(121,170)(122,169)(123,172)(124,171)(125,174)
(126,173)(127,176)(128,175)(129,178)(130,177)(131,180)(132,179)(133,158)
(134,157)(135,160)(136,159)(137,162)(138,161)(139,164)(140,163)(141,166)
(142,165)(143,168)(144,167)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192)(193,206)(194,205)(195,208)(196,207)(197,210)(198,209)(199,212)
(200,211)(201,214)(202,213)(203,216)(204,215)(217,254)(218,253)(219,256)
(220,255)(221,258)(222,257)(223,260)(224,259)(225,262)(226,261)(227,264)
(228,263)(229,278)(230,277)(231,280)(232,279)(233,282)(234,281)(235,284)
(236,283)(237,286)(238,285)(239,288)(240,287)(241,266)(242,265)(243,268)
(244,267)(245,270)(246,269)(247,272)(248,271)(249,274)(250,273)(251,276)
(252,275)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,314)
(302,313)(303,316)(304,315)(305,318)(306,317)(307,320)(308,319)(309,322)
(310,321)(311,324)(312,323);
poly := sub<Sym(324)|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*s0*s1*s2*s0*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*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 >;

```
References : None.
to this polytope