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Polytope of Type {4,18,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18,3}*1296
if this polytope has a name.
Group : SmallGroup(1296,943)
Rank : 4
Schlafli Type : {4,18,3}
Number of vertices, edges, etc : 4, 108, 81, 9
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
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 : {2,18,3}*648
   3-fold quotients : {4,6,3}*432a
   6-fold quotients : {2,6,3}*216
   9-fold quotients : {4,6,3}*144
   18-fold quotients : {2,6,3}*72
   27-fold quotients : {4,2,3}*48
   54-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,163)(  2,164)(  3,165)(  4,166)(  5,167)(  6,168)(  7,169)(  8,170)
(  9,171)( 10,172)( 11,173)( 12,174)( 13,175)( 14,176)( 15,177)( 16,178)
( 17,179)( 18,180)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)( 24,186)
( 25,187)( 26,188)( 27,189)( 28,190)( 29,191)( 30,192)( 31,193)( 32,194)
( 33,195)( 34,196)( 35,197)( 36,198)( 37,199)( 38,200)( 39,201)( 40,202)
( 41,203)( 42,204)( 43,205)( 44,206)( 45,207)( 46,208)( 47,209)( 48,210)
( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,217)( 56,218)
( 57,219)( 58,220)( 59,221)( 60,222)( 61,223)( 62,224)( 63,225)( 64,226)
( 65,227)( 66,228)( 67,229)( 68,230)( 69,231)( 70,232)( 71,233)( 72,234)
( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)( 80,242)
( 81,243)( 82,244)( 83,245)( 84,246)( 85,247)( 86,248)( 87,249)( 88,250)
( 89,251)( 90,252)( 91,253)( 92,254)( 93,255)( 94,256)( 95,257)( 96,258)
( 97,259)( 98,260)( 99,261)(100,262)(101,263)(102,264)(103,265)(104,266)
(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)(112,274)
(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)(120,282)
(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)(128,290)
(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,298)
(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)(144,306)
(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)(152,314)
(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)
(161,323)(162,324);;
s1 := (  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)(163,244)(164,246)
(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)(172,253)
(173,255)(174,254)(175,259)(176,261)(177,260)(178,256)(179,258)(180,257)
(181,262)(182,264)(183,263)(184,268)(185,270)(186,269)(187,265)(188,267)
(189,266)(190,300)(191,299)(192,298)(193,306)(194,305)(195,304)(196,303)
(197,302)(198,301)(199,309)(200,308)(201,307)(202,315)(203,314)(204,313)
(205,312)(206,311)(207,310)(208,318)(209,317)(210,316)(211,324)(212,323)
(213,322)(214,321)(215,320)(216,319)(217,273)(218,272)(219,271)(220,279)
(221,278)(222,277)(223,276)(224,275)(225,274)(226,282)(227,281)(228,280)
(229,288)(230,287)(231,286)(232,285)(233,284)(234,283)(235,291)(236,290)
(237,289)(238,297)(239,296)(240,295)(241,294)(242,293)(243,292);;
s2 := (  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)(163,190)(164,192)
(165,191)(166,195)(167,194)(168,193)(169,197)(170,196)(171,198)(172,215)
(173,214)(174,216)(175,208)(176,210)(177,209)(178,213)(179,212)(180,211)
(181,202)(182,204)(183,203)(184,207)(185,206)(186,205)(187,200)(188,199)
(189,201)(217,219)(220,221)(224,225)(226,241)(227,243)(228,242)(229,237)
(230,236)(231,235)(232,239)(233,238)(234,240)(244,271)(245,273)(246,272)
(247,276)(248,275)(249,274)(250,278)(251,277)(252,279)(253,296)(254,295)
(255,297)(256,289)(257,291)(258,290)(259,294)(260,293)(261,292)(262,283)
(263,285)(264,284)(265,288)(266,287)(267,286)(268,281)(269,280)(270,282)
(298,300)(301,302)(305,306)(307,322)(308,324)(309,323)(310,318)(311,317)
(312,316)(313,320)(314,319)(315,321);;
s3 := (  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)(  8, 18)
(  9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 66)( 29, 65)( 30, 64)( 31, 69)
( 32, 68)( 33, 67)( 34, 72)( 35, 71)( 36, 70)( 37, 57)( 38, 56)( 39, 55)
( 40, 60)( 41, 59)( 42, 58)( 43, 63)( 44, 62)( 45, 61)( 46, 75)( 47, 74)
( 48, 73)( 49, 78)( 50, 77)( 51, 76)( 52, 81)( 53, 80)( 54, 79)( 82, 91)
( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)
(101,102)(104,105)(107,108)(109,147)(110,146)(111,145)(112,150)(113,149)
(114,148)(115,153)(116,152)(117,151)(118,138)(119,137)(120,136)(121,141)
(122,140)(123,139)(124,144)(125,143)(126,142)(127,156)(128,155)(129,154)
(130,159)(131,158)(132,157)(133,162)(134,161)(135,160)(163,172)(164,174)
(165,173)(166,175)(167,177)(168,176)(169,178)(170,180)(171,179)(182,183)
(185,186)(188,189)(190,228)(191,227)(192,226)(193,231)(194,230)(195,229)
(196,234)(197,233)(198,232)(199,219)(200,218)(201,217)(202,222)(203,221)
(204,220)(205,225)(206,224)(207,223)(208,237)(209,236)(210,235)(211,240)
(212,239)(213,238)(214,243)(215,242)(216,241)(244,253)(245,255)(246,254)
(247,256)(248,258)(249,257)(250,259)(251,261)(252,260)(263,264)(266,267)
(269,270)(271,309)(272,308)(273,307)(274,312)(275,311)(276,310)(277,315)
(278,314)(279,313)(280,300)(281,299)(282,298)(283,303)(284,302)(285,301)
(286,306)(287,305)(288,304)(289,318)(290,317)(291,316)(292,321)(293,320)
(294,319)(295,324)(296,323)(297,322);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(324)!(  1,163)(  2,164)(  3,165)(  4,166)(  5,167)(  6,168)(  7,169)
(  8,170)(  9,171)( 10,172)( 11,173)( 12,174)( 13,175)( 14,176)( 15,177)
( 16,178)( 17,179)( 18,180)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)
( 24,186)( 25,187)( 26,188)( 27,189)( 28,190)( 29,191)( 30,192)( 31,193)
( 32,194)( 33,195)( 34,196)( 35,197)( 36,198)( 37,199)( 38,200)( 39,201)
( 40,202)( 41,203)( 42,204)( 43,205)( 44,206)( 45,207)( 46,208)( 47,209)
( 48,210)( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,217)
( 56,218)( 57,219)( 58,220)( 59,221)( 60,222)( 61,223)( 62,224)( 63,225)
( 64,226)( 65,227)( 66,228)( 67,229)( 68,230)( 69,231)( 70,232)( 71,233)
( 72,234)( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)
( 80,242)( 81,243)( 82,244)( 83,245)( 84,246)( 85,247)( 86,248)( 87,249)
( 88,250)( 89,251)( 90,252)( 91,253)( 92,254)( 93,255)( 94,256)( 95,257)
( 96,258)( 97,259)( 98,260)( 99,261)(100,262)(101,263)(102,264)(103,265)
(104,266)(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)
(112,274)(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)
(120,282)(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)
(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)
(136,298)(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)
(144,306)(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)
(152,314)(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)
(160,322)(161,323)(162,324);
s1 := 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)(163,244)
(164,246)(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)
(172,253)(173,255)(174,254)(175,259)(176,261)(177,260)(178,256)(179,258)
(180,257)(181,262)(182,264)(183,263)(184,268)(185,270)(186,269)(187,265)
(188,267)(189,266)(190,300)(191,299)(192,298)(193,306)(194,305)(195,304)
(196,303)(197,302)(198,301)(199,309)(200,308)(201,307)(202,315)(203,314)
(204,313)(205,312)(206,311)(207,310)(208,318)(209,317)(210,316)(211,324)
(212,323)(213,322)(214,321)(215,320)(216,319)(217,273)(218,272)(219,271)
(220,279)(221,278)(222,277)(223,276)(224,275)(225,274)(226,282)(227,281)
(228,280)(229,288)(230,287)(231,286)(232,285)(233,284)(234,283)(235,291)
(236,290)(237,289)(238,297)(239,296)(240,295)(241,294)(242,293)(243,292);
s2 := 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)(163,190)
(164,192)(165,191)(166,195)(167,194)(168,193)(169,197)(170,196)(171,198)
(172,215)(173,214)(174,216)(175,208)(176,210)(177,209)(178,213)(179,212)
(180,211)(181,202)(182,204)(183,203)(184,207)(185,206)(186,205)(187,200)
(188,199)(189,201)(217,219)(220,221)(224,225)(226,241)(227,243)(228,242)
(229,237)(230,236)(231,235)(232,239)(233,238)(234,240)(244,271)(245,273)
(246,272)(247,276)(248,275)(249,274)(250,278)(251,277)(252,279)(253,296)
(254,295)(255,297)(256,289)(257,291)(258,290)(259,294)(260,293)(261,292)
(262,283)(263,285)(264,284)(265,288)(266,287)(267,286)(268,281)(269,280)
(270,282)(298,300)(301,302)(305,306)(307,322)(308,324)(309,323)(310,318)
(311,317)(312,316)(313,320)(314,319)(315,321);
s3 := Sym(324)!(  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)
(  8, 18)(  9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 66)( 29, 65)( 30, 64)
( 31, 69)( 32, 68)( 33, 67)( 34, 72)( 35, 71)( 36, 70)( 37, 57)( 38, 56)
( 39, 55)( 40, 60)( 41, 59)( 42, 58)( 43, 63)( 44, 62)( 45, 61)( 46, 75)
( 47, 74)( 48, 73)( 49, 78)( 50, 77)( 51, 76)( 52, 81)( 53, 80)( 54, 79)
( 82, 91)( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)
( 90, 98)(101,102)(104,105)(107,108)(109,147)(110,146)(111,145)(112,150)
(113,149)(114,148)(115,153)(116,152)(117,151)(118,138)(119,137)(120,136)
(121,141)(122,140)(123,139)(124,144)(125,143)(126,142)(127,156)(128,155)
(129,154)(130,159)(131,158)(132,157)(133,162)(134,161)(135,160)(163,172)
(164,174)(165,173)(166,175)(167,177)(168,176)(169,178)(170,180)(171,179)
(182,183)(185,186)(188,189)(190,228)(191,227)(192,226)(193,231)(194,230)
(195,229)(196,234)(197,233)(198,232)(199,219)(200,218)(201,217)(202,222)
(203,221)(204,220)(205,225)(206,224)(207,223)(208,237)(209,236)(210,235)
(211,240)(212,239)(213,238)(214,243)(215,242)(216,241)(244,253)(245,255)
(246,254)(247,256)(248,258)(249,257)(250,259)(251,261)(252,260)(263,264)
(266,267)(269,270)(271,309)(272,308)(273,307)(274,312)(275,311)(276,310)
(277,315)(278,314)(279,313)(280,300)(281,299)(282,298)(283,303)(284,302)
(285,301)(286,306)(287,305)(288,304)(289,318)(290,317)(291,316)(292,321)
(293,320)(294,319)(295,324)(296,323)(297,322);
poly := sub<Sym(324)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
to this polytope