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Polytope of Type {18,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6,6}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1858)
Rank : 4
Schlafli Type : {18,6,6}
Number of vertices, edges, etc : 18, 54, 18, 6
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 6
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 : {9,6,6}*648a, {18,6,3}*648a
   3-fold quotients : {18,2,6}*432, {6,6,6}*432a
   4-fold quotients : {9,6,3}*324
   6-fold quotients : {9,2,6}*216, {18,2,3}*216, {3,6,6}*216a, {6,6,3}*216a
   9-fold quotients : {18,2,2}*144, {6,2,6}*144
   12-fold quotients : {9,2,3}*108, {3,6,3}*108
   18-fold quotients : {9,2,2}*72, {3,2,6}*72, {6,2,3}*72
   27-fold quotients : {2,2,6}*48, {6,2,2}*48
   36-fold quotients : {3,2,3}*36
   54-fold quotients : {2,2,3}*24, {3,2,2}*24
   81-fold quotients : {2,2,2}*16
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, 58)( 29, 60)( 30, 59)( 31, 55)
( 32, 57)( 33, 56)( 34, 61)( 35, 63)( 36, 62)( 37, 67)( 38, 69)( 39, 68)
( 40, 64)( 41, 66)( 42, 65)( 43, 70)( 44, 72)( 45, 71)( 46, 76)( 47, 78)
( 48, 77)( 49, 73)( 50, 75)( 51, 74)( 52, 79)( 53, 81)( 54, 80)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,139)(110,141)(111,140)(112,136)(113,138)
(114,137)(115,142)(116,144)(117,143)(118,148)(119,150)(120,149)(121,145)
(122,147)(123,146)(124,151)(125,153)(126,152)(127,157)(128,159)(129,158)
(130,154)(131,156)(132,155)(133,160)(134,162)(135,161)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,220)(191,222)(192,221)(193,217)(194,219)(195,218)
(196,223)(197,225)(198,224)(199,229)(200,231)(201,230)(202,226)(203,228)
(204,227)(205,232)(206,234)(207,233)(208,238)(209,240)(210,239)(211,235)
(212,237)(213,236)(214,241)(215,243)(216,242)(245,246)(247,250)(248,252)
(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)(266,270)
(267,269)(271,301)(272,303)(273,302)(274,298)(275,300)(276,299)(277,304)
(278,306)(279,305)(280,310)(281,312)(282,311)(283,307)(284,309)(285,308)
(286,313)(287,315)(288,314)(289,319)(290,321)(291,320)(292,316)(293,318)
(294,317)(295,322)(296,324)(297,323);;
s1 := (  1,190)(  2,192)(  3,191)(  4,196)(  5,198)(  6,197)(  7,193)(  8,195)
(  9,194)( 10,200)( 11,199)( 12,201)( 13,206)( 14,205)( 15,207)( 16,203)
( 17,202)( 18,204)( 19,210)( 20,209)( 21,208)( 22,216)( 23,215)( 24,214)
( 25,213)( 26,212)( 27,211)( 28,163)( 29,165)( 30,164)( 31,169)( 32,171)
( 33,170)( 34,166)( 35,168)( 36,167)( 37,173)( 38,172)( 39,174)( 40,179)
( 41,178)( 42,180)( 43,176)( 44,175)( 45,177)( 46,183)( 47,182)( 48,181)
( 49,189)( 50,188)( 51,187)( 52,186)( 53,185)( 54,184)( 55,220)( 56,222)
( 57,221)( 58,217)( 59,219)( 60,218)( 61,223)( 62,225)( 63,224)( 64,230)
( 65,229)( 66,231)( 67,227)( 68,226)( 69,228)( 70,233)( 71,232)( 72,234)
( 73,240)( 74,239)( 75,238)( 76,237)( 77,236)( 78,235)( 79,243)( 80,242)
( 81,241)( 82,271)( 83,273)( 84,272)( 85,277)( 86,279)( 87,278)( 88,274)
( 89,276)( 90,275)( 91,281)( 92,280)( 93,282)( 94,287)( 95,286)( 96,288)
( 97,284)( 98,283)( 99,285)(100,291)(101,290)(102,289)(103,297)(104,296)
(105,295)(106,294)(107,293)(108,292)(109,244)(110,246)(111,245)(112,250)
(113,252)(114,251)(115,247)(116,249)(117,248)(118,254)(119,253)(120,255)
(121,260)(122,259)(123,261)(124,257)(125,256)(126,258)(127,264)(128,263)
(129,262)(130,270)(131,269)(132,268)(133,267)(134,266)(135,265)(136,301)
(137,303)(138,302)(139,298)(140,300)(141,299)(142,304)(143,306)(144,305)
(145,311)(146,310)(147,312)(148,308)(149,307)(150,309)(151,314)(152,313)
(153,315)(154,321)(155,320)(156,319)(157,318)(158,317)(159,316)(160,324)
(161,323)(162,322);;
s2 := (  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, 37)( 29, 39)( 30, 38)( 31, 40)
( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 47, 48)( 50, 51)( 53, 54)
( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)( 62, 72)
( 63, 71)( 74, 75)( 77, 78)( 80, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)
( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)(107,108)
(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)(116,126)
(117,125)(128,129)(131,132)(134,135)(136,145)(137,147)(138,146)(139,148)
(140,150)(141,149)(142,151)(143,153)(144,152)(155,156)(158,159)(161,162)
(163,172)(164,174)(165,173)(166,175)(167,177)(168,176)(169,178)(170,180)
(171,179)(182,183)(185,186)(188,189)(190,199)(191,201)(192,200)(193,202)
(194,204)(195,203)(196,205)(197,207)(198,206)(209,210)(212,213)(215,216)
(217,226)(218,228)(219,227)(220,229)(221,231)(222,230)(223,232)(224,234)
(225,233)(236,237)(239,240)(242,243)(244,253)(245,255)(246,254)(247,256)
(248,258)(249,257)(250,259)(251,261)(252,260)(263,264)(266,267)(269,270)
(271,280)(272,282)(273,281)(274,283)(275,285)(276,284)(277,286)(278,288)
(279,287)(290,291)(293,294)(296,297)(298,307)(299,309)(300,308)(301,310)
(302,312)(303,311)(304,313)(305,315)(306,314)(317,318)(320,321)(323,324);;
s3 := (  1, 82)(  2, 84)(  3, 83)(  4, 85)(  5, 87)(  6, 86)(  7, 88)(  8, 90)
(  9, 89)( 10,100)( 11,102)( 12,101)( 13,103)( 14,105)( 15,104)( 16,106)
( 17,108)( 18,107)( 19, 91)( 20, 93)( 21, 92)( 22, 94)( 23, 96)( 24, 95)
( 25, 97)( 26, 99)( 27, 98)( 28,109)( 29,111)( 30,110)( 31,112)( 32,114)
( 33,113)( 34,115)( 35,117)( 36,116)( 37,127)( 38,129)( 39,128)( 40,130)
( 41,132)( 42,131)( 43,133)( 44,135)( 45,134)( 46,118)( 47,120)( 48,119)
( 49,121)( 50,123)( 51,122)( 52,124)( 53,126)( 54,125)( 55,136)( 56,138)
( 57,137)( 58,139)( 59,141)( 60,140)( 61,142)( 62,144)( 63,143)( 64,154)
( 65,156)( 66,155)( 67,157)( 68,159)( 69,158)( 70,160)( 71,162)( 72,161)
( 73,145)( 74,147)( 75,146)( 76,148)( 77,150)( 78,149)( 79,151)( 80,153)
( 81,152)(163,244)(164,246)(165,245)(166,247)(167,249)(168,248)(169,250)
(170,252)(171,251)(172,262)(173,264)(174,263)(175,265)(176,267)(177,266)
(178,268)(179,270)(180,269)(181,253)(182,255)(183,254)(184,256)(185,258)
(186,257)(187,259)(188,261)(189,260)(190,271)(191,273)(192,272)(193,274)
(194,276)(195,275)(196,277)(197,279)(198,278)(199,289)(200,291)(201,290)
(202,292)(203,294)(204,293)(205,295)(206,297)(207,296)(208,280)(209,282)
(210,281)(211,283)(212,285)(213,284)(214,286)(215,288)(216,287)(217,298)
(218,300)(219,299)(220,301)(221,303)(222,302)(223,304)(224,306)(225,305)
(226,316)(227,318)(228,317)(229,319)(230,321)(231,320)(232,322)(233,324)
(234,323)(235,307)(236,309)(237,308)(238,310)(239,312)(240,311)(241,313)
(242,315)(243,314);;
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*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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, 58)( 29, 60)( 30, 59)
( 31, 55)( 32, 57)( 33, 56)( 34, 61)( 35, 63)( 36, 62)( 37, 67)( 38, 69)
( 39, 68)( 40, 64)( 41, 66)( 42, 65)( 43, 70)( 44, 72)( 45, 71)( 46, 76)
( 47, 78)( 48, 77)( 49, 73)( 50, 75)( 51, 74)( 52, 79)( 53, 81)( 54, 80)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,139)(110,141)(111,140)(112,136)
(113,138)(114,137)(115,142)(116,144)(117,143)(118,148)(119,150)(120,149)
(121,145)(122,147)(123,146)(124,151)(125,153)(126,152)(127,157)(128,159)
(129,158)(130,154)(131,156)(132,155)(133,160)(134,162)(135,161)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(190,220)(191,222)(192,221)(193,217)(194,219)
(195,218)(196,223)(197,225)(198,224)(199,229)(200,231)(201,230)(202,226)
(203,228)(204,227)(205,232)(206,234)(207,233)(208,238)(209,240)(210,239)
(211,235)(212,237)(213,236)(214,241)(215,243)(216,242)(245,246)(247,250)
(248,252)(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)
(266,270)(267,269)(271,301)(272,303)(273,302)(274,298)(275,300)(276,299)
(277,304)(278,306)(279,305)(280,310)(281,312)(282,311)(283,307)(284,309)
(285,308)(286,313)(287,315)(288,314)(289,319)(290,321)(291,320)(292,316)
(293,318)(294,317)(295,322)(296,324)(297,323);
s1 := Sym(324)!(  1,190)(  2,192)(  3,191)(  4,196)(  5,198)(  6,197)(  7,193)
(  8,195)(  9,194)( 10,200)( 11,199)( 12,201)( 13,206)( 14,205)( 15,207)
( 16,203)( 17,202)( 18,204)( 19,210)( 20,209)( 21,208)( 22,216)( 23,215)
( 24,214)( 25,213)( 26,212)( 27,211)( 28,163)( 29,165)( 30,164)( 31,169)
( 32,171)( 33,170)( 34,166)( 35,168)( 36,167)( 37,173)( 38,172)( 39,174)
( 40,179)( 41,178)( 42,180)( 43,176)( 44,175)( 45,177)( 46,183)( 47,182)
( 48,181)( 49,189)( 50,188)( 51,187)( 52,186)( 53,185)( 54,184)( 55,220)
( 56,222)( 57,221)( 58,217)( 59,219)( 60,218)( 61,223)( 62,225)( 63,224)
( 64,230)( 65,229)( 66,231)( 67,227)( 68,226)( 69,228)( 70,233)( 71,232)
( 72,234)( 73,240)( 74,239)( 75,238)( 76,237)( 77,236)( 78,235)( 79,243)
( 80,242)( 81,241)( 82,271)( 83,273)( 84,272)( 85,277)( 86,279)( 87,278)
( 88,274)( 89,276)( 90,275)( 91,281)( 92,280)( 93,282)( 94,287)( 95,286)
( 96,288)( 97,284)( 98,283)( 99,285)(100,291)(101,290)(102,289)(103,297)
(104,296)(105,295)(106,294)(107,293)(108,292)(109,244)(110,246)(111,245)
(112,250)(113,252)(114,251)(115,247)(116,249)(117,248)(118,254)(119,253)
(120,255)(121,260)(122,259)(123,261)(124,257)(125,256)(126,258)(127,264)
(128,263)(129,262)(130,270)(131,269)(132,268)(133,267)(134,266)(135,265)
(136,301)(137,303)(138,302)(139,298)(140,300)(141,299)(142,304)(143,306)
(144,305)(145,311)(146,310)(147,312)(148,308)(149,307)(150,309)(151,314)
(152,313)(153,315)(154,321)(155,320)(156,319)(157,318)(158,317)(159,316)
(160,324)(161,323)(162,322);
s2 := 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, 37)( 29, 39)( 30, 38)
( 31, 40)( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 47, 48)( 50, 51)
( 53, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)
( 62, 72)( 63, 71)( 74, 75)( 77, 78)( 80, 81)( 82, 91)( 83, 93)( 84, 92)
( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)
(107,108)(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)
(116,126)(117,125)(128,129)(131,132)(134,135)(136,145)(137,147)(138,146)
(139,148)(140,150)(141,149)(142,151)(143,153)(144,152)(155,156)(158,159)
(161,162)(163,172)(164,174)(165,173)(166,175)(167,177)(168,176)(169,178)
(170,180)(171,179)(182,183)(185,186)(188,189)(190,199)(191,201)(192,200)
(193,202)(194,204)(195,203)(196,205)(197,207)(198,206)(209,210)(212,213)
(215,216)(217,226)(218,228)(219,227)(220,229)(221,231)(222,230)(223,232)
(224,234)(225,233)(236,237)(239,240)(242,243)(244,253)(245,255)(246,254)
(247,256)(248,258)(249,257)(250,259)(251,261)(252,260)(263,264)(266,267)
(269,270)(271,280)(272,282)(273,281)(274,283)(275,285)(276,284)(277,286)
(278,288)(279,287)(290,291)(293,294)(296,297)(298,307)(299,309)(300,308)
(301,310)(302,312)(303,311)(304,313)(305,315)(306,314)(317,318)(320,321)
(323,324);
s3 := Sym(324)!(  1, 82)(  2, 84)(  3, 83)(  4, 85)(  5, 87)(  6, 86)(  7, 88)
(  8, 90)(  9, 89)( 10,100)( 11,102)( 12,101)( 13,103)( 14,105)( 15,104)
( 16,106)( 17,108)( 18,107)( 19, 91)( 20, 93)( 21, 92)( 22, 94)( 23, 96)
( 24, 95)( 25, 97)( 26, 99)( 27, 98)( 28,109)( 29,111)( 30,110)( 31,112)
( 32,114)( 33,113)( 34,115)( 35,117)( 36,116)( 37,127)( 38,129)( 39,128)
( 40,130)( 41,132)( 42,131)( 43,133)( 44,135)( 45,134)( 46,118)( 47,120)
( 48,119)( 49,121)( 50,123)( 51,122)( 52,124)( 53,126)( 54,125)( 55,136)
( 56,138)( 57,137)( 58,139)( 59,141)( 60,140)( 61,142)( 62,144)( 63,143)
( 64,154)( 65,156)( 66,155)( 67,157)( 68,159)( 69,158)( 70,160)( 71,162)
( 72,161)( 73,145)( 74,147)( 75,146)( 76,148)( 77,150)( 78,149)( 79,151)
( 80,153)( 81,152)(163,244)(164,246)(165,245)(166,247)(167,249)(168,248)
(169,250)(170,252)(171,251)(172,262)(173,264)(174,263)(175,265)(176,267)
(177,266)(178,268)(179,270)(180,269)(181,253)(182,255)(183,254)(184,256)
(185,258)(186,257)(187,259)(188,261)(189,260)(190,271)(191,273)(192,272)
(193,274)(194,276)(195,275)(196,277)(197,279)(198,278)(199,289)(200,291)
(201,290)(202,292)(203,294)(204,293)(205,295)(206,297)(207,296)(208,280)
(209,282)(210,281)(211,283)(212,285)(213,284)(214,286)(215,288)(216,287)
(217,298)(218,300)(219,299)(220,301)(221,303)(222,302)(223,304)(224,306)
(225,305)(226,316)(227,318)(228,317)(229,319)(230,321)(231,320)(232,322)
(233,324)(234,323)(235,307)(236,309)(237,308)(238,310)(239,312)(240,311)
(241,313)(242,315)(243,314);
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*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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