Polytope of Type {18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12}*1296d
if this polytope has a name.
Group : SmallGroup(1296,867)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 54, 324, 36
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 18
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 : {18,6}*648h
   3-fold quotients : {6,12}*432a
   4-fold quotients : {18,3}*324
   6-fold quotients : {6,6}*216a
   9-fold quotients : {6,12}*144b
   12-fold quotients : {6,3}*108
   18-fold quotients : {6,6}*72b
   27-fold quotients : {2,12}*48
   36-fold quotients : {6,3}*36
   54-fold quotients : {2,6}*24
   81-fold quotients : {2,4}*16
   108-fold quotients : {2,3}*12
   162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope