Polytope of Type {18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12}*1296c
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 : 6
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}*648g
   3-fold quotients : {6,12}*432b
   4-fold quotients : {18,6}*324c
   6-fold quotients : {6,6}*216b
   9-fold quotients : {6,12}*144a
   12-fold quotients : {6,6}*108
   18-fold quotients : {6,6}*72a
   27-fold quotients : {2,12}*48, {6,4}*48a
   54-fold quotients : {2,6}*24, {6,2}*24
   81-fold quotients : {2,4}*16
   108-fold quotients : {2,3}*12, {3,2}*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)(  2,173)(  3,174)(  4,178)(  5,179)(  6,180)(  7,175)(  8,176)
(  9,177)( 10,163)( 11,164)( 12,165)( 13,169)( 14,170)( 15,171)( 16,166)
( 17,167)( 18,168)( 19,181)( 20,182)( 21,183)( 22,187)( 23,188)( 24,189)
( 25,184)( 26,185)( 27,186)( 28,199)( 29,200)( 30,201)( 31,205)( 32,206)
( 33,207)( 34,202)( 35,203)( 36,204)( 37,190)( 38,191)( 39,192)( 40,196)
( 41,197)( 42,198)( 43,193)( 44,194)( 45,195)( 46,208)( 47,209)( 48,210)
( 49,214)( 50,215)( 51,216)( 52,211)( 53,212)( 54,213)( 55,226)( 56,227)
( 57,228)( 58,232)( 59,233)( 60,234)( 61,229)( 62,230)( 63,231)( 64,217)
( 65,218)( 66,219)( 67,223)( 68,224)( 69,225)( 70,220)( 71,221)( 72,222)
( 73,235)( 74,236)( 75,237)( 76,241)( 77,242)( 78,243)( 79,238)( 80,239)
( 81,240)( 82,253)( 83,254)( 84,255)( 85,259)( 86,260)( 87,261)( 88,256)
( 89,257)( 90,258)( 91,244)( 92,245)( 93,246)( 94,250)( 95,251)( 96,252)
( 97,247)( 98,248)( 99,249)(100,262)(101,263)(102,264)(103,268)(104,269)
(105,270)(106,265)(107,266)(108,267)(109,280)(110,281)(111,282)(112,286)
(113,287)(114,288)(115,283)(116,284)(117,285)(118,271)(119,272)(120,273)
(121,277)(122,278)(123,279)(124,274)(125,275)(126,276)(127,289)(128,290)
(129,291)(130,295)(131,296)(132,297)(133,292)(134,293)(135,294)(136,307)
(137,308)(138,309)(139,313)(140,314)(141,315)(142,310)(143,311)(144,312)
(145,298)(146,299)(147,300)(148,304)(149,305)(150,306)(151,301)(152,302)
(153,303)(154,316)(155,317)(156,318)(157,322)(158,323)(159,324)(160,319)
(161,320)(162,321);;
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, s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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 ];;
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)(  2,173)(  3,174)(  4,178)(  5,179)(  6,180)(  7,175)
(  8,176)(  9,177)( 10,163)( 11,164)( 12,165)( 13,169)( 14,170)( 15,171)
( 16,166)( 17,167)( 18,168)( 19,181)( 20,182)( 21,183)( 22,187)( 23,188)
( 24,189)( 25,184)( 26,185)( 27,186)( 28,199)( 29,200)( 30,201)( 31,205)
( 32,206)( 33,207)( 34,202)( 35,203)( 36,204)( 37,190)( 38,191)( 39,192)
( 40,196)( 41,197)( 42,198)( 43,193)( 44,194)( 45,195)( 46,208)( 47,209)
( 48,210)( 49,214)( 50,215)( 51,216)( 52,211)( 53,212)( 54,213)( 55,226)
( 56,227)( 57,228)( 58,232)( 59,233)( 60,234)( 61,229)( 62,230)( 63,231)
( 64,217)( 65,218)( 66,219)( 67,223)( 68,224)( 69,225)( 70,220)( 71,221)
( 72,222)( 73,235)( 74,236)( 75,237)( 76,241)( 77,242)( 78,243)( 79,238)
( 80,239)( 81,240)( 82,253)( 83,254)( 84,255)( 85,259)( 86,260)( 87,261)
( 88,256)( 89,257)( 90,258)( 91,244)( 92,245)( 93,246)( 94,250)( 95,251)
( 96,252)( 97,247)( 98,248)( 99,249)(100,262)(101,263)(102,264)(103,268)
(104,269)(105,270)(106,265)(107,266)(108,267)(109,280)(110,281)(111,282)
(112,286)(113,287)(114,288)(115,283)(116,284)(117,285)(118,271)(119,272)
(120,273)(121,277)(122,278)(123,279)(124,274)(125,275)(126,276)(127,289)
(128,290)(129,291)(130,295)(131,296)(132,297)(133,292)(134,293)(135,294)
(136,307)(137,308)(138,309)(139,313)(140,314)(141,315)(142,310)(143,311)
(144,312)(145,298)(146,299)(147,300)(148,304)(149,305)(150,306)(151,301)
(152,302)(153,303)(154,316)(155,317)(156,318)(157,322)(158,323)(159,324)
(160,319)(161,320)(162,321);
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, s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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 >;

References : None.
to this polytope