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Polytope of Type {9,18,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,18,4}*1296
if this polytope has a name.
Group : SmallGroup(1296,878)
Rank : 4
Schlafli Type : {9,18,4}
Number of vertices, edges, etc : 9, 81, 36, 4
Order of s0s1s2s3 : 36
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 : {9,18,2}*648
   3-fold quotients : {9,6,4}*432
   6-fold quotients : {9,6,2}*216
   9-fold quotients : {9,2,4}*144, {3,6,4}*144
   18-fold quotients : {9,2,2}*72, {3,6,2}*72
   27-fold quotients : {3,2,4}*48
   54-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 21)( 11, 20)( 12, 19)( 13, 27)
( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 28, 61)( 29, 63)( 30, 62)
( 31, 58)( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 81)( 38, 80)
( 39, 79)( 40, 78)( 41, 77)( 42, 76)( 43, 75)( 44, 74)( 45, 73)( 46, 72)
( 47, 71)( 48, 70)( 49, 69)( 50, 68)( 51, 67)( 52, 66)( 53, 65)( 54, 64)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 91,102)( 92,101)( 93,100)( 94,108)
( 95,107)( 96,106)( 97,105)( 98,104)( 99,103)(109,142)(110,144)(111,143)
(112,139)(113,141)(114,140)(115,136)(116,138)(117,137)(118,162)(119,161)
(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)(127,153)
(128,152)(129,151)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145)
(164,165)(166,169)(167,171)(168,170)(172,183)(173,182)(174,181)(175,189)
(176,188)(177,187)(178,186)(179,185)(180,184)(190,223)(191,225)(192,224)
(193,220)(194,222)(195,221)(196,217)(197,219)(198,218)(199,243)(200,242)
(201,241)(202,240)(203,239)(204,238)(205,237)(206,236)(207,235)(208,234)
(209,233)(210,232)(211,231)(212,230)(213,229)(214,228)(215,227)(216,226)
(245,246)(247,250)(248,252)(249,251)(253,264)(254,263)(255,262)(256,270)
(257,269)(258,268)(259,267)(260,266)(261,265)(271,304)(272,306)(273,305)
(274,301)(275,303)(276,302)(277,298)(278,300)(279,299)(280,324)(281,323)
(282,322)(283,321)(284,320)(285,319)(286,318)(287,317)(288,316)(289,315)
(290,314)(291,313)(292,312)(293,311)(294,310)(295,309)(296,308)(297,307);;
s1 := (  1, 37)(  2, 39)(  3, 38)(  4, 43)(  5, 45)(  6, 44)(  7, 40)(  8, 42)
(  9, 41)( 10, 28)( 11, 30)( 12, 29)( 13, 34)( 14, 36)( 15, 35)( 16, 31)
( 17, 33)( 18, 32)( 19, 48)( 20, 47)( 21, 46)( 22, 54)( 23, 53)( 24, 52)
( 25, 51)( 26, 50)( 27, 49)( 55, 70)( 56, 72)( 57, 71)( 58, 67)( 59, 69)
( 60, 68)( 61, 64)( 62, 66)( 63, 65)( 73, 81)( 74, 80)( 75, 79)( 76, 78)
( 82,118)( 83,120)( 84,119)( 85,124)( 86,126)( 87,125)( 88,121)( 89,123)
( 90,122)( 91,109)( 92,111)( 93,110)( 94,115)( 95,117)( 96,116)( 97,112)
( 98,114)( 99,113)(100,129)(101,128)(102,127)(103,135)(104,134)(105,133)
(106,132)(107,131)(108,130)(136,151)(137,153)(138,152)(139,148)(140,150)
(141,149)(142,145)(143,147)(144,146)(154,162)(155,161)(156,160)(157,159)
(163,199)(164,201)(165,200)(166,205)(167,207)(168,206)(169,202)(170,204)
(171,203)(172,190)(173,192)(174,191)(175,196)(176,198)(177,197)(178,193)
(179,195)(180,194)(181,210)(182,209)(183,208)(184,216)(185,215)(186,214)
(187,213)(188,212)(189,211)(217,232)(218,234)(219,233)(220,229)(221,231)
(222,230)(223,226)(224,228)(225,227)(235,243)(236,242)(237,241)(238,240)
(244,280)(245,282)(246,281)(247,286)(248,288)(249,287)(250,283)(251,285)
(252,284)(253,271)(254,273)(255,272)(256,277)(257,279)(258,278)(259,274)
(260,276)(261,275)(262,291)(263,290)(264,289)(265,297)(266,296)(267,295)
(268,294)(269,293)(270,292)(298,313)(299,315)(300,314)(301,310)(302,312)
(303,311)(304,307)(305,309)(306,308)(316,324)(317,323)(318,322)(319,321);;
s2 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 28, 61)( 29, 62)( 30, 63)( 31, 58)( 32, 59)( 33, 60)( 34, 55)
( 35, 56)( 36, 57)( 37, 70)( 38, 71)( 39, 72)( 40, 67)( 41, 68)( 42, 69)
( 43, 64)( 44, 65)( 45, 66)( 46, 79)( 47, 80)( 48, 81)( 49, 76)( 50, 77)
( 51, 78)( 52, 73)( 53, 74)( 54, 75)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,142)(110,143)(111,144)
(112,139)(113,140)(114,141)(115,136)(116,137)(117,138)(118,151)(119,152)
(120,153)(121,148)(122,149)(123,150)(124,145)(125,146)(126,147)(127,160)
(128,161)(129,162)(130,157)(131,158)(132,159)(133,154)(134,155)(135,156)
(163,244)(164,245)(165,246)(166,250)(167,251)(168,252)(169,247)(170,248)
(171,249)(172,253)(173,254)(174,255)(175,259)(176,260)(177,261)(178,256)
(179,257)(180,258)(181,262)(182,263)(183,264)(184,268)(185,269)(186,270)
(187,265)(188,266)(189,267)(190,304)(191,305)(192,306)(193,301)(194,302)
(195,303)(196,298)(197,299)(198,300)(199,313)(200,314)(201,315)(202,310)
(203,311)(204,312)(205,307)(206,308)(207,309)(208,322)(209,323)(210,324)
(211,319)(212,320)(213,321)(214,316)(215,317)(216,318)(217,277)(218,278)
(219,279)(220,274)(221,275)(222,276)(223,271)(224,272)(225,273)(226,286)
(227,287)(228,288)(229,283)(230,284)(231,285)(232,280)(233,281)(234,282)
(235,295)(236,296)(237,297)(238,292)(239,293)(240,294)(241,289)(242,290)
(243,291);;
s3 := (  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);;
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*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)( 10, 21)( 11, 20)( 12, 19)
( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 28, 61)( 29, 63)
( 30, 62)( 31, 58)( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 81)
( 38, 80)( 39, 79)( 40, 78)( 41, 77)( 42, 76)( 43, 75)( 44, 74)( 45, 73)
( 46, 72)( 47, 71)( 48, 70)( 49, 69)( 50, 68)( 51, 67)( 52, 66)( 53, 65)
( 54, 64)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 91,102)( 92,101)( 93,100)
( 94,108)( 95,107)( 96,106)( 97,105)( 98,104)( 99,103)(109,142)(110,144)
(111,143)(112,139)(113,141)(114,140)(115,136)(116,138)(117,137)(118,162)
(119,161)(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)
(127,153)(128,152)(129,151)(130,150)(131,149)(132,148)(133,147)(134,146)
(135,145)(164,165)(166,169)(167,171)(168,170)(172,183)(173,182)(174,181)
(175,189)(176,188)(177,187)(178,186)(179,185)(180,184)(190,223)(191,225)
(192,224)(193,220)(194,222)(195,221)(196,217)(197,219)(198,218)(199,243)
(200,242)(201,241)(202,240)(203,239)(204,238)(205,237)(206,236)(207,235)
(208,234)(209,233)(210,232)(211,231)(212,230)(213,229)(214,228)(215,227)
(216,226)(245,246)(247,250)(248,252)(249,251)(253,264)(254,263)(255,262)
(256,270)(257,269)(258,268)(259,267)(260,266)(261,265)(271,304)(272,306)
(273,305)(274,301)(275,303)(276,302)(277,298)(278,300)(279,299)(280,324)
(281,323)(282,322)(283,321)(284,320)(285,319)(286,318)(287,317)(288,316)
(289,315)(290,314)(291,313)(292,312)(293,311)(294,310)(295,309)(296,308)
(297,307);
s1 := Sym(324)!(  1, 37)(  2, 39)(  3, 38)(  4, 43)(  5, 45)(  6, 44)(  7, 40)
(  8, 42)(  9, 41)( 10, 28)( 11, 30)( 12, 29)( 13, 34)( 14, 36)( 15, 35)
( 16, 31)( 17, 33)( 18, 32)( 19, 48)( 20, 47)( 21, 46)( 22, 54)( 23, 53)
( 24, 52)( 25, 51)( 26, 50)( 27, 49)( 55, 70)( 56, 72)( 57, 71)( 58, 67)
( 59, 69)( 60, 68)( 61, 64)( 62, 66)( 63, 65)( 73, 81)( 74, 80)( 75, 79)
( 76, 78)( 82,118)( 83,120)( 84,119)( 85,124)( 86,126)( 87,125)( 88,121)
( 89,123)( 90,122)( 91,109)( 92,111)( 93,110)( 94,115)( 95,117)( 96,116)
( 97,112)( 98,114)( 99,113)(100,129)(101,128)(102,127)(103,135)(104,134)
(105,133)(106,132)(107,131)(108,130)(136,151)(137,153)(138,152)(139,148)
(140,150)(141,149)(142,145)(143,147)(144,146)(154,162)(155,161)(156,160)
(157,159)(163,199)(164,201)(165,200)(166,205)(167,207)(168,206)(169,202)
(170,204)(171,203)(172,190)(173,192)(174,191)(175,196)(176,198)(177,197)
(178,193)(179,195)(180,194)(181,210)(182,209)(183,208)(184,216)(185,215)
(186,214)(187,213)(188,212)(189,211)(217,232)(218,234)(219,233)(220,229)
(221,231)(222,230)(223,226)(224,228)(225,227)(235,243)(236,242)(237,241)
(238,240)(244,280)(245,282)(246,281)(247,286)(248,288)(249,287)(250,283)
(251,285)(252,284)(253,271)(254,273)(255,272)(256,277)(257,279)(258,278)
(259,274)(260,276)(261,275)(262,291)(263,290)(264,289)(265,297)(266,296)
(267,295)(268,294)(269,293)(270,292)(298,313)(299,315)(300,314)(301,310)
(302,312)(303,311)(304,307)(305,309)(306,308)(316,324)(317,323)(318,322)
(319,321);
s2 := Sym(324)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 28, 61)( 29, 62)( 30, 63)( 31, 58)( 32, 59)( 33, 60)
( 34, 55)( 35, 56)( 36, 57)( 37, 70)( 38, 71)( 39, 72)( 40, 67)( 41, 68)
( 42, 69)( 43, 64)( 44, 65)( 45, 66)( 46, 79)( 47, 80)( 48, 81)( 49, 76)
( 50, 77)( 51, 78)( 52, 73)( 53, 74)( 54, 75)( 85, 88)( 86, 89)( 87, 90)
( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,142)(110,143)
(111,144)(112,139)(113,140)(114,141)(115,136)(116,137)(117,138)(118,151)
(119,152)(120,153)(121,148)(122,149)(123,150)(124,145)(125,146)(126,147)
(127,160)(128,161)(129,162)(130,157)(131,158)(132,159)(133,154)(134,155)
(135,156)(163,244)(164,245)(165,246)(166,250)(167,251)(168,252)(169,247)
(170,248)(171,249)(172,253)(173,254)(174,255)(175,259)(176,260)(177,261)
(178,256)(179,257)(180,258)(181,262)(182,263)(183,264)(184,268)(185,269)
(186,270)(187,265)(188,266)(189,267)(190,304)(191,305)(192,306)(193,301)
(194,302)(195,303)(196,298)(197,299)(198,300)(199,313)(200,314)(201,315)
(202,310)(203,311)(204,312)(205,307)(206,308)(207,309)(208,322)(209,323)
(210,324)(211,319)(212,320)(213,321)(214,316)(215,317)(216,318)(217,277)
(218,278)(219,279)(220,274)(221,275)(222,276)(223,271)(224,272)(225,273)
(226,286)(227,287)(228,288)(229,283)(230,284)(231,285)(232,280)(233,281)
(234,282)(235,295)(236,296)(237,297)(238,292)(239,293)(240,294)(241,289)
(242,290)(243,291);
s3 := 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);
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope