Polytope of Type {18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12}*1296a
if this polytope has a name.
Group : SmallGroup(1296,811)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 54, 324, 36
Order of s₀s₁s₂ : 36
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}*648b
   3-fold quotients : {18,12}*432a, {6,12}*432b
   4-fold quotients : {18,6}*324a
   6-fold quotients : {18,6}*216a, {6,6}*216b
   9-fold quotients : {18,4}*144a, {6,12}*144a
   12-fold quotients : {6,6}*108
   18-fold quotients : {18,2}*72, {6,6}*72a
   27-fold quotients : {2,12}*48, {6,4}*48a
   36-fold quotients : {9,2}*36
   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, 61)( 29, 63)( 30, 62)( 31, 58)
( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 70)( 38, 72)( 39, 71)
( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 46, 79)( 47, 81)
( 48, 80)( 49, 76)( 50, 78)( 51, 77)( 52, 73)( 53, 75)( 54, 74)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,142)(110,144)(111,143)(112,139)(113,141)
(114,140)(115,136)(116,138)(117,137)(118,151)(119,153)(120,152)(121,148)
(122,150)(123,149)(124,145)(125,147)(126,146)(127,160)(128,162)(129,161)
(130,157)(131,159)(132,158)(133,154)(134,156)(135,155)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,223)(191,225)(192,224)(193,220)(194,222)(195,221)
(196,217)(197,219)(198,218)(199,232)(200,234)(201,233)(202,229)(203,231)
(204,230)(205,226)(206,228)(207,227)(208,241)(209,243)(210,242)(211,238)
(212,240)(213,239)(214,235)(215,237)(216,236)(245,246)(247,250)(248,252)
(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)(266,270)
(267,269)(271,304)(272,306)(273,305)(274,301)(275,303)(276,302)(277,298)
(278,300)(279,299)(280,313)(281,315)(282,314)(283,310)(284,312)(285,311)
(286,307)(287,309)(288,308)(289,322)(290,324)(291,323)(292,319)(293,321)
(294,320)(295,316)(296,318)(297,317);;
s1 := (  1, 28)(  2, 29)(  3, 30)(  4, 34)(  5, 35)(  6, 36)(  7, 31)(  8, 32)
(  9, 33)( 10, 48)( 11, 46)( 12, 47)( 13, 54)( 14, 52)( 15, 53)( 16, 51)
( 17, 49)( 18, 50)( 19, 38)( 20, 39)( 21, 37)( 22, 44)( 23, 45)( 24, 43)
( 25, 41)( 26, 42)( 27, 40)( 55, 61)( 56, 62)( 57, 63)( 64, 81)( 65, 79)
( 66, 80)( 67, 78)( 68, 76)( 69, 77)( 70, 75)( 71, 73)( 72, 74)( 82,109)
( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)
( 91,129)( 92,127)( 93,128)( 94,135)( 95,133)( 96,134)( 97,132)( 98,130)
( 99,131)(100,119)(101,120)(102,118)(103,125)(104,126)(105,124)(106,122)
(107,123)(108,121)(136,142)(137,143)(138,144)(145,162)(146,160)(147,161)
(148,159)(149,157)(150,158)(151,156)(152,154)(153,155)(163,271)(164,272)
(165,273)(166,277)(167,278)(168,279)(169,274)(170,275)(171,276)(172,291)
(173,289)(174,290)(175,297)(176,295)(177,296)(178,294)(179,292)(180,293)
(181,281)(182,282)(183,280)(184,287)(185,288)(186,286)(187,284)(188,285)
(189,283)(190,244)(191,245)(192,246)(193,250)(194,251)(195,252)(196,247)
(197,248)(198,249)(199,264)(200,262)(201,263)(202,270)(203,268)(204,269)
(205,267)(206,265)(207,266)(208,254)(209,255)(210,253)(211,260)(212,261)
(213,259)(214,257)(215,258)(216,256)(217,304)(218,305)(219,306)(220,301)
(221,302)(222,303)(223,298)(224,299)(225,300)(226,324)(227,322)(228,323)
(229,321)(230,319)(231,320)(232,318)(233,316)(234,317)(235,314)(236,315)
(237,313)(238,311)(239,312)(240,310)(241,308)(242,309)(243,307);;
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)( 28,199)( 29,201)( 30,200)( 31,202)( 32,204)
( 33,203)( 34,205)( 35,207)( 36,206)( 37,190)( 38,192)( 39,191)( 40,193)
( 41,195)( 42,194)( 43,196)( 44,198)( 45,197)( 46,208)( 47,210)( 48,209)
( 49,211)( 50,213)( 51,212)( 52,214)( 53,216)( 54,215)( 55,226)( 56,228)
( 57,227)( 58,229)( 59,231)( 60,230)( 61,232)( 62,234)( 63,233)( 64,217)
( 65,219)( 66,218)( 67,220)( 68,222)( 69,221)( 70,223)( 71,225)( 72,224)
( 73,235)( 74,237)( 75,236)( 76,238)( 77,240)( 78,239)( 79,241)( 80,243)
( 81,242)( 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)(109,280)(110,282)(111,281)(112,283)
(113,285)(114,284)(115,286)(116,288)(117,287)(118,271)(119,273)(120,272)
(121,274)(122,276)(123,275)(124,277)(125,279)(126,278)(127,289)(128,291)
(129,290)(130,292)(131,294)(132,293)(133,295)(134,297)(135,296)(136,307)
(137,309)(138,308)(139,310)(140,312)(141,311)(142,313)(143,315)(144,314)
(145,298)(146,300)(147,299)(148,301)(149,303)(150,302)(151,304)(152,306)
(153,305)(154,316)(155,318)(156,317)(157,319)(158,321)(159,320)(160,322)
(161,324)(162,323);;
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*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*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, 
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(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, 61)( 29, 63)( 30, 62)
( 31, 58)( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 70)( 38, 72)
( 39, 71)( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 46, 79)
( 47, 81)( 48, 80)( 49, 76)( 50, 78)( 51, 77)( 52, 73)( 53, 75)( 54, 74)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,142)(110,144)(111,143)(112,139)
(113,141)(114,140)(115,136)(116,138)(117,137)(118,151)(119,153)(120,152)
(121,148)(122,150)(123,149)(124,145)(125,147)(126,146)(127,160)(128,162)
(129,161)(130,157)(131,159)(132,158)(133,154)(134,156)(135,155)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(190,223)(191,225)(192,224)(193,220)(194,222)
(195,221)(196,217)(197,219)(198,218)(199,232)(200,234)(201,233)(202,229)
(203,231)(204,230)(205,226)(206,228)(207,227)(208,241)(209,243)(210,242)
(211,238)(212,240)(213,239)(214,235)(215,237)(216,236)(245,246)(247,250)
(248,252)(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)
(266,270)(267,269)(271,304)(272,306)(273,305)(274,301)(275,303)(276,302)
(277,298)(278,300)(279,299)(280,313)(281,315)(282,314)(283,310)(284,312)
(285,311)(286,307)(287,309)(288,308)(289,322)(290,324)(291,323)(292,319)
(293,321)(294,320)(295,316)(296,318)(297,317);
s1 := Sym(324)!(  1, 28)(  2, 29)(  3, 30)(  4, 34)(  5, 35)(  6, 36)(  7, 31)
(  8, 32)(  9, 33)( 10, 48)( 11, 46)( 12, 47)( 13, 54)( 14, 52)( 15, 53)
( 16, 51)( 17, 49)( 18, 50)( 19, 38)( 20, 39)( 21, 37)( 22, 44)( 23, 45)
( 24, 43)( 25, 41)( 26, 42)( 27, 40)( 55, 61)( 56, 62)( 57, 63)( 64, 81)
( 65, 79)( 66, 80)( 67, 78)( 68, 76)( 69, 77)( 70, 75)( 71, 73)( 72, 74)
( 82,109)( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)
( 90,114)( 91,129)( 92,127)( 93,128)( 94,135)( 95,133)( 96,134)( 97,132)
( 98,130)( 99,131)(100,119)(101,120)(102,118)(103,125)(104,126)(105,124)
(106,122)(107,123)(108,121)(136,142)(137,143)(138,144)(145,162)(146,160)
(147,161)(148,159)(149,157)(150,158)(151,156)(152,154)(153,155)(163,271)
(164,272)(165,273)(166,277)(167,278)(168,279)(169,274)(170,275)(171,276)
(172,291)(173,289)(174,290)(175,297)(176,295)(177,296)(178,294)(179,292)
(180,293)(181,281)(182,282)(183,280)(184,287)(185,288)(186,286)(187,284)
(188,285)(189,283)(190,244)(191,245)(192,246)(193,250)(194,251)(195,252)
(196,247)(197,248)(198,249)(199,264)(200,262)(201,263)(202,270)(203,268)
(204,269)(205,267)(206,265)(207,266)(208,254)(209,255)(210,253)(211,260)
(212,261)(213,259)(214,257)(215,258)(216,256)(217,304)(218,305)(219,306)
(220,301)(221,302)(222,303)(223,298)(224,299)(225,300)(226,324)(227,322)
(228,323)(229,321)(230,319)(231,320)(232,318)(233,316)(234,317)(235,314)
(236,315)(237,313)(238,311)(239,312)(240,310)(241,308)(242,309)(243,307);
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)( 28,199)( 29,201)( 30,200)( 31,202)
( 32,204)( 33,203)( 34,205)( 35,207)( 36,206)( 37,190)( 38,192)( 39,191)
( 40,193)( 41,195)( 42,194)( 43,196)( 44,198)( 45,197)( 46,208)( 47,210)
( 48,209)( 49,211)( 50,213)( 51,212)( 52,214)( 53,216)( 54,215)( 55,226)
( 56,228)( 57,227)( 58,229)( 59,231)( 60,230)( 61,232)( 62,234)( 63,233)
( 64,217)( 65,219)( 66,218)( 67,220)( 68,222)( 69,221)( 70,223)( 71,225)
( 72,224)( 73,235)( 74,237)( 75,236)( 76,238)( 77,240)( 78,239)( 79,241)
( 80,243)( 81,242)( 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)(109,280)(110,282)(111,281)
(112,283)(113,285)(114,284)(115,286)(116,288)(117,287)(118,271)(119,273)
(120,272)(121,274)(122,276)(123,275)(124,277)(125,279)(126,278)(127,289)
(128,291)(129,290)(130,292)(131,294)(132,293)(133,295)(134,297)(135,296)
(136,307)(137,309)(138,308)(139,310)(140,312)(141,311)(142,313)(143,315)
(144,314)(145,298)(146,300)(147,299)(148,301)(149,303)(150,302)(151,304)
(152,306)(153,305)(154,316)(155,318)(156,317)(157,319)(158,321)(159,320)
(160,322)(161,324)(162,323);
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*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*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, 
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