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Polytope of Type {6,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,18}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1858)
Rank : 4
Schlafli Type : {6,6,18}
Number of vertices, edges, etc : 6, 18, 54, 18
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 : {3,6,18}*648a, {6,6,9}*648a
   3-fold quotients : {6,2,18}*432, {6,6,6}*432a
   4-fold quotients : {3,6,9}*324
   6-fold quotients : {3,2,18}*216, {6,2,9}*216, {3,6,6}*216a, {6,6,3}*216a
   9-fold quotients : {2,2,18}*144, {6,2,6}*144
   12-fold quotients : {3,2,9}*108, {3,6,3}*108
   18-fold quotients : {2,2,9}*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)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)
(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)
(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161)
(164,165)(167,168)(170,171)(172,181)(173,183)(174,182)(175,184)(176,186)
(177,185)(178,187)(179,189)(180,188)(191,192)(194,195)(197,198)(199,208)
(200,210)(201,209)(202,211)(203,213)(204,212)(205,214)(206,216)(207,215)
(218,219)(221,222)(224,225)(226,235)(227,237)(228,236)(229,238)(230,240)
(231,239)(232,241)(233,243)(234,242)(245,246)(248,249)(251,252)(253,262)
(254,264)(255,263)(256,265)(257,267)(258,266)(259,268)(260,270)(261,269)
(272,273)(275,276)(278,279)(280,289)(281,291)(282,290)(283,292)(284,294)
(285,293)(286,295)(287,297)(288,296)(299,300)(302,303)(305,306)(307,316)
(308,318)(309,317)(310,319)(311,321)(312,320)(313,322)(314,324)(315,323);;
s1 := (  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);;
s2 := (  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)(  8, 33)
(  9, 32)( 10, 38)( 11, 37)( 12, 39)( 13, 44)( 14, 43)( 15, 45)( 16, 41)
( 17, 40)( 18, 42)( 19, 48)( 20, 47)( 21, 46)( 22, 54)( 23, 53)( 24, 52)
( 25, 51)( 26, 50)( 27, 49)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 68)
( 65, 67)( 66, 69)( 70, 71)( 73, 78)( 74, 77)( 75, 76)( 79, 81)( 82,109)
( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)( 90,113)
( 91,119)( 92,118)( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)( 98,121)
( 99,123)(100,129)(101,128)(102,127)(103,135)(104,134)(105,133)(106,132)
(107,131)(108,130)(136,139)(137,141)(138,140)(143,144)(145,149)(146,148)
(147,150)(151,152)(154,159)(155,158)(156,157)(160,162)(163,190)(164,192)
(165,191)(166,196)(167,198)(168,197)(169,193)(170,195)(171,194)(172,200)
(173,199)(174,201)(175,206)(176,205)(177,207)(178,203)(179,202)(180,204)
(181,210)(182,209)(183,208)(184,216)(185,215)(186,214)(187,213)(188,212)
(189,211)(217,220)(218,222)(219,221)(224,225)(226,230)(227,229)(228,231)
(232,233)(235,240)(236,239)(237,238)(241,243)(244,271)(245,273)(246,272)
(247,277)(248,279)(249,278)(250,274)(251,276)(252,275)(253,281)(254,280)
(255,282)(256,287)(257,286)(258,288)(259,284)(260,283)(261,285)(262,291)
(263,290)(264,289)(265,297)(266,296)(267,295)(268,294)(269,293)(270,292)
(298,301)(299,303)(300,302)(305,306)(307,311)(308,310)(309,312)(313,314)
(316,321)(317,320)(318,319)(322,324);;
s3 := (  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)(  8, 87)
(  9, 86)( 10, 91)( 11, 93)( 12, 92)( 13, 97)( 14, 99)( 15, 98)( 16, 94)
( 17, 96)( 18, 95)( 19,100)( 20,102)( 21,101)( 22,106)( 23,108)( 24,107)
( 25,103)( 26,105)( 27,104)( 28,139)( 29,141)( 30,140)( 31,136)( 32,138)
( 33,137)( 34,142)( 35,144)( 36,143)( 37,148)( 38,150)( 39,149)( 40,145)
( 41,147)( 42,146)( 43,151)( 44,153)( 45,152)( 46,157)( 47,159)( 48,158)
( 49,154)( 50,156)( 51,155)( 52,160)( 53,162)( 54,161)( 55,112)( 56,114)
( 57,113)( 58,109)( 59,111)( 60,110)( 61,115)( 62,117)( 63,116)( 64,121)
( 65,123)( 66,122)( 67,118)( 68,120)( 69,119)( 70,124)( 71,126)( 72,125)
( 73,130)( 74,132)( 75,131)( 76,127)( 77,129)( 78,128)( 79,133)( 80,135)
( 81,134)(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)(190,301)(191,303)(192,302)(193,298)
(194,300)(195,299)(196,304)(197,306)(198,305)(199,310)(200,312)(201,311)
(202,307)(203,309)(204,308)(205,313)(206,315)(207,314)(208,319)(209,321)
(210,320)(211,316)(212,318)(213,317)(214,322)(215,324)(216,323)(217,274)
(218,276)(219,275)(220,271)(221,273)(222,272)(223,277)(224,279)(225,278)
(226,283)(227,285)(228,284)(229,280)(230,282)(231,281)(232,286)(233,288)
(234,287)(235,292)(236,294)(237,293)(238,289)(239,291)(240,290)(241,295)
(242,297)(243,296);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(324)!(  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)
( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)
( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)
( 99,107)(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)
(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)
(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)
(153,161)(164,165)(167,168)(170,171)(172,181)(173,183)(174,182)(175,184)
(176,186)(177,185)(178,187)(179,189)(180,188)(191,192)(194,195)(197,198)
(199,208)(200,210)(201,209)(202,211)(203,213)(204,212)(205,214)(206,216)
(207,215)(218,219)(221,222)(224,225)(226,235)(227,237)(228,236)(229,238)
(230,240)(231,239)(232,241)(233,243)(234,242)(245,246)(248,249)(251,252)
(253,262)(254,264)(255,263)(256,265)(257,267)(258,266)(259,268)(260,270)
(261,269)(272,273)(275,276)(278,279)(280,289)(281,291)(282,290)(283,292)
(284,294)(285,293)(286,295)(287,297)(288,296)(299,300)(302,303)(305,306)
(307,316)(308,318)(309,317)(310,319)(311,321)(312,320)(313,322)(314,324)
(315,323);
s1 := 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);
s2 := Sym(324)!(  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)
(  8, 33)(  9, 32)( 10, 38)( 11, 37)( 12, 39)( 13, 44)( 14, 43)( 15, 45)
( 16, 41)( 17, 40)( 18, 42)( 19, 48)( 20, 47)( 21, 46)( 22, 54)( 23, 53)
( 24, 52)( 25, 51)( 26, 50)( 27, 49)( 55, 58)( 56, 60)( 57, 59)( 62, 63)
( 64, 68)( 65, 67)( 66, 69)( 70, 71)( 73, 78)( 74, 77)( 75, 76)( 79, 81)
( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)
( 90,113)( 91,119)( 92,118)( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)
( 98,121)( 99,123)(100,129)(101,128)(102,127)(103,135)(104,134)(105,133)
(106,132)(107,131)(108,130)(136,139)(137,141)(138,140)(143,144)(145,149)
(146,148)(147,150)(151,152)(154,159)(155,158)(156,157)(160,162)(163,190)
(164,192)(165,191)(166,196)(167,198)(168,197)(169,193)(170,195)(171,194)
(172,200)(173,199)(174,201)(175,206)(176,205)(177,207)(178,203)(179,202)
(180,204)(181,210)(182,209)(183,208)(184,216)(185,215)(186,214)(187,213)
(188,212)(189,211)(217,220)(218,222)(219,221)(224,225)(226,230)(227,229)
(228,231)(232,233)(235,240)(236,239)(237,238)(241,243)(244,271)(245,273)
(246,272)(247,277)(248,279)(249,278)(250,274)(251,276)(252,275)(253,281)
(254,280)(255,282)(256,287)(257,286)(258,288)(259,284)(260,283)(261,285)
(262,291)(263,290)(264,289)(265,297)(266,296)(267,295)(268,294)(269,293)
(270,292)(298,301)(299,303)(300,302)(305,306)(307,311)(308,310)(309,312)
(313,314)(316,321)(317,320)(318,319)(322,324);
s3 := Sym(324)!(  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)
(  8, 87)(  9, 86)( 10, 91)( 11, 93)( 12, 92)( 13, 97)( 14, 99)( 15, 98)
( 16, 94)( 17, 96)( 18, 95)( 19,100)( 20,102)( 21,101)( 22,106)( 23,108)
( 24,107)( 25,103)( 26,105)( 27,104)( 28,139)( 29,141)( 30,140)( 31,136)
( 32,138)( 33,137)( 34,142)( 35,144)( 36,143)( 37,148)( 38,150)( 39,149)
( 40,145)( 41,147)( 42,146)( 43,151)( 44,153)( 45,152)( 46,157)( 47,159)
( 48,158)( 49,154)( 50,156)( 51,155)( 52,160)( 53,162)( 54,161)( 55,112)
( 56,114)( 57,113)( 58,109)( 59,111)( 60,110)( 61,115)( 62,117)( 63,116)
( 64,121)( 65,123)( 66,122)( 67,118)( 68,120)( 69,119)( 70,124)( 71,126)
( 72,125)( 73,130)( 74,132)( 75,131)( 76,127)( 77,129)( 78,128)( 79,133)
( 80,135)( 81,134)(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)(190,301)(191,303)(192,302)
(193,298)(194,300)(195,299)(196,304)(197,306)(198,305)(199,310)(200,312)
(201,311)(202,307)(203,309)(204,308)(205,313)(206,315)(207,314)(208,319)
(209,321)(210,320)(211,316)(212,318)(213,317)(214,322)(215,324)(216,323)
(217,274)(218,276)(219,275)(220,271)(221,273)(222,272)(223,277)(224,279)
(225,278)(226,283)(227,285)(228,284)(229,280)(230,282)(231,281)(232,286)
(233,288)(234,287)(235,292)(236,294)(237,293)(238,289)(239,291)(240,290)
(241,295)(242,297)(243,296);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
to this polytope