Polytope of Type {18,9,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,9,4}*1296
if this polytope has a name.
Group : SmallGroup(1296,1782)
Rank : 4
Schlafli Type : {18,9,4}
Number of vertices, edges, etc : 18, 81, 18, 4
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-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) :
3-fold quotients : {6,9,4}*432
9-fold quotients : {2,9,4}*144, {6,3,4}*144
27-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)
( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 37, 97)( 38, 98)( 39, 99)( 40,100)
( 41,101)( 42,102)( 43,103)( 44,104)( 45,105)( 46,106)( 47,107)( 48,108)
( 49, 85)( 50, 86)( 51, 87)( 52, 88)( 53, 89)( 54, 90)( 55, 91)( 56, 92)
( 57, 93)( 58, 94)( 59, 95)( 60, 96)( 61, 73)( 62, 74)( 63, 75)( 64, 76)
( 65, 77)( 66, 78)( 67, 79)( 68, 80)( 69, 81)( 70, 82)( 71, 83)( 72, 84)
(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)(128,140)
(129,141)(130,142)(131,143)(132,144)(145,205)(146,206)(147,207)(148,208)
(149,209)(150,210)(151,211)(152,212)(153,213)(154,214)(155,215)(156,216)
(157,193)(158,194)(159,195)(160,196)(161,197)(162,198)(163,199)(164,200)
(165,201)(166,202)(167,203)(168,204)(169,181)(170,182)(171,183)(172,184)
(173,185)(174,186)(175,187)(176,188)(177,189)(178,190)(179,191)(180,192)
(229,241)(230,242)(231,243)(232,244)(233,245)(234,246)(235,247)(236,248)
(237,249)(238,250)(239,251)(240,252)(253,313)(254,314)(255,315)(256,316)
(257,317)(258,318)(259,319)(260,320)(261,321)(262,322)(263,323)(264,324)
(265,301)(266,302)(267,303)(268,304)(269,305)(270,306)(271,307)(272,308)
(273,309)(274,310)(275,311)(276,312)(277,289)(278,290)(279,291)(280,292)
(281,293)(282,294)(283,295)(284,296)(285,297)(286,298)(287,299)(288,300);;
s1 := ( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 45)( 6, 46)( 7, 48)( 8, 47)
( 9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 61)( 14, 62)( 15, 64)( 16, 63)
( 17, 69)( 18, 70)( 19, 72)( 20, 71)( 21, 65)( 22, 66)( 23, 68)( 24, 67)
( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 57)( 30, 58)( 31, 60)( 32, 59)
( 33, 53)( 34, 54)( 35, 56)( 36, 55)( 73, 97)( 74, 98)( 75,100)( 76, 99)
( 77,105)( 78,106)( 79,108)( 80,107)( 81,101)( 82,102)( 83,104)( 84,103)
( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)(109,261)(110,262)(111,264)
(112,263)(113,257)(114,258)(115,260)(116,259)(117,253)(118,254)(119,256)
(120,255)(121,285)(122,286)(123,288)(124,287)(125,281)(126,282)(127,284)
(128,283)(129,277)(130,278)(131,280)(132,279)(133,273)(134,274)(135,276)
(136,275)(137,269)(138,270)(139,272)(140,271)(141,265)(142,266)(143,268)
(144,267)(145,225)(146,226)(147,228)(148,227)(149,221)(150,222)(151,224)
(152,223)(153,217)(154,218)(155,220)(156,219)(157,249)(158,250)(159,252)
(160,251)(161,245)(162,246)(163,248)(164,247)(165,241)(166,242)(167,244)
(168,243)(169,237)(170,238)(171,240)(172,239)(173,233)(174,234)(175,236)
(176,235)(177,229)(178,230)(179,232)(180,231)(181,321)(182,322)(183,324)
(184,323)(185,317)(186,318)(187,320)(188,319)(189,313)(190,314)(191,316)
(192,315)(193,309)(194,310)(195,312)(196,311)(197,305)(198,306)(199,308)
(200,307)(201,301)(202,302)(203,304)(204,303)(205,297)(206,298)(207,300)
(208,299)(209,293)(210,294)(211,296)(212,295)(213,289)(214,290)(215,292)
(216,291);;
s2 := ( 1,109)( 2,112)( 3,111)( 4,110)( 5,117)( 6,120)( 7,119)( 8,118)
( 9,113)( 10,116)( 11,115)( 12,114)( 13,133)( 14,136)( 15,135)( 16,134)
( 17,141)( 18,144)( 19,143)( 20,142)( 21,137)( 22,140)( 23,139)( 24,138)
( 25,121)( 26,124)( 27,123)( 28,122)( 29,129)( 30,132)( 31,131)( 32,130)
( 33,125)( 34,128)( 35,127)( 36,126)( 37,205)( 38,208)( 39,207)( 40,206)
( 41,213)( 42,216)( 43,215)( 44,214)( 45,209)( 46,212)( 47,211)( 48,210)
( 49,193)( 50,196)( 51,195)( 52,194)( 53,201)( 54,204)( 55,203)( 56,202)
( 57,197)( 58,200)( 59,199)( 60,198)( 61,181)( 62,184)( 63,183)( 64,182)
( 65,189)( 66,192)( 67,191)( 68,190)( 69,185)( 70,188)( 71,187)( 72,186)
( 73,169)( 74,172)( 75,171)( 76,170)( 77,177)( 78,180)( 79,179)( 80,178)
( 81,173)( 82,176)( 83,175)( 84,174)( 85,157)( 86,160)( 87,159)( 88,158)
( 89,165)( 90,168)( 91,167)( 92,166)( 93,161)( 94,164)( 95,163)( 96,162)
( 97,145)( 98,148)( 99,147)(100,146)(101,153)(102,156)(103,155)(104,154)
(105,149)(106,152)(107,151)(108,150)(217,225)(218,228)(219,227)(220,226)
(222,224)(229,249)(230,252)(231,251)(232,250)(233,245)(234,248)(235,247)
(236,246)(237,241)(238,244)(239,243)(240,242)(253,321)(254,324)(255,323)
(256,322)(257,317)(258,320)(259,319)(260,318)(261,313)(262,316)(263,315)
(264,314)(265,309)(266,312)(267,311)(268,310)(269,305)(270,308)(271,307)
(272,306)(273,301)(274,304)(275,303)(276,302)(277,297)(278,300)(279,299)
(280,298)(281,293)(282,296)(283,295)(284,294)(285,289)(286,292)(287,291)
(288,290);;
s3 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)
(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)
(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)
(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)
(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)
(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)
(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)
(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304)
(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)(319,320)
(321,322)(323,324);;
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, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s2*s1*s3*s2*s3*s2*s1*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(324)!( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)
( 20, 32)( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 37, 97)( 38, 98)( 39, 99)
( 40,100)( 41,101)( 42,102)( 43,103)( 44,104)( 45,105)( 46,106)( 47,107)
( 48,108)( 49, 85)( 50, 86)( 51, 87)( 52, 88)( 53, 89)( 54, 90)( 55, 91)
( 56, 92)( 57, 93)( 58, 94)( 59, 95)( 60, 96)( 61, 73)( 62, 74)( 63, 75)
( 64, 76)( 65, 77)( 66, 78)( 67, 79)( 68, 80)( 69, 81)( 70, 82)( 71, 83)
( 72, 84)(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)
(128,140)(129,141)(130,142)(131,143)(132,144)(145,205)(146,206)(147,207)
(148,208)(149,209)(150,210)(151,211)(152,212)(153,213)(154,214)(155,215)
(156,216)(157,193)(158,194)(159,195)(160,196)(161,197)(162,198)(163,199)
(164,200)(165,201)(166,202)(167,203)(168,204)(169,181)(170,182)(171,183)
(172,184)(173,185)(174,186)(175,187)(176,188)(177,189)(178,190)(179,191)
(180,192)(229,241)(230,242)(231,243)(232,244)(233,245)(234,246)(235,247)
(236,248)(237,249)(238,250)(239,251)(240,252)(253,313)(254,314)(255,315)
(256,316)(257,317)(258,318)(259,319)(260,320)(261,321)(262,322)(263,323)
(264,324)(265,301)(266,302)(267,303)(268,304)(269,305)(270,306)(271,307)
(272,308)(273,309)(274,310)(275,311)(276,312)(277,289)(278,290)(279,291)
(280,292)(281,293)(282,294)(283,295)(284,296)(285,297)(286,298)(287,299)
(288,300);
s1 := Sym(324)!( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 45)( 6, 46)( 7, 48)
( 8, 47)( 9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 61)( 14, 62)( 15, 64)
( 16, 63)( 17, 69)( 18, 70)( 19, 72)( 20, 71)( 21, 65)( 22, 66)( 23, 68)
( 24, 67)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 57)( 30, 58)( 31, 60)
( 32, 59)( 33, 53)( 34, 54)( 35, 56)( 36, 55)( 73, 97)( 74, 98)( 75,100)
( 76, 99)( 77,105)( 78,106)( 79,108)( 80,107)( 81,101)( 82,102)( 83,104)
( 84,103)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)(109,261)(110,262)
(111,264)(112,263)(113,257)(114,258)(115,260)(116,259)(117,253)(118,254)
(119,256)(120,255)(121,285)(122,286)(123,288)(124,287)(125,281)(126,282)
(127,284)(128,283)(129,277)(130,278)(131,280)(132,279)(133,273)(134,274)
(135,276)(136,275)(137,269)(138,270)(139,272)(140,271)(141,265)(142,266)
(143,268)(144,267)(145,225)(146,226)(147,228)(148,227)(149,221)(150,222)
(151,224)(152,223)(153,217)(154,218)(155,220)(156,219)(157,249)(158,250)
(159,252)(160,251)(161,245)(162,246)(163,248)(164,247)(165,241)(166,242)
(167,244)(168,243)(169,237)(170,238)(171,240)(172,239)(173,233)(174,234)
(175,236)(176,235)(177,229)(178,230)(179,232)(180,231)(181,321)(182,322)
(183,324)(184,323)(185,317)(186,318)(187,320)(188,319)(189,313)(190,314)
(191,316)(192,315)(193,309)(194,310)(195,312)(196,311)(197,305)(198,306)
(199,308)(200,307)(201,301)(202,302)(203,304)(204,303)(205,297)(206,298)
(207,300)(208,299)(209,293)(210,294)(211,296)(212,295)(213,289)(214,290)
(215,292)(216,291);
s2 := Sym(324)!( 1,109)( 2,112)( 3,111)( 4,110)( 5,117)( 6,120)( 7,119)
( 8,118)( 9,113)( 10,116)( 11,115)( 12,114)( 13,133)( 14,136)( 15,135)
( 16,134)( 17,141)( 18,144)( 19,143)( 20,142)( 21,137)( 22,140)( 23,139)
( 24,138)( 25,121)( 26,124)( 27,123)( 28,122)( 29,129)( 30,132)( 31,131)
( 32,130)( 33,125)( 34,128)( 35,127)( 36,126)( 37,205)( 38,208)( 39,207)
( 40,206)( 41,213)( 42,216)( 43,215)( 44,214)( 45,209)( 46,212)( 47,211)
( 48,210)( 49,193)( 50,196)( 51,195)( 52,194)( 53,201)( 54,204)( 55,203)
( 56,202)( 57,197)( 58,200)( 59,199)( 60,198)( 61,181)( 62,184)( 63,183)
( 64,182)( 65,189)( 66,192)( 67,191)( 68,190)( 69,185)( 70,188)( 71,187)
( 72,186)( 73,169)( 74,172)( 75,171)( 76,170)( 77,177)( 78,180)( 79,179)
( 80,178)( 81,173)( 82,176)( 83,175)( 84,174)( 85,157)( 86,160)( 87,159)
( 88,158)( 89,165)( 90,168)( 91,167)( 92,166)( 93,161)( 94,164)( 95,163)
( 96,162)( 97,145)( 98,148)( 99,147)(100,146)(101,153)(102,156)(103,155)
(104,154)(105,149)(106,152)(107,151)(108,150)(217,225)(218,228)(219,227)
(220,226)(222,224)(229,249)(230,252)(231,251)(232,250)(233,245)(234,248)
(235,247)(236,246)(237,241)(238,244)(239,243)(240,242)(253,321)(254,324)
(255,323)(256,322)(257,317)(258,320)(259,319)(260,318)(261,313)(262,316)
(263,315)(264,314)(265,309)(266,312)(267,311)(268,310)(269,305)(270,308)
(271,307)(272,306)(273,301)(274,304)(275,303)(276,302)(277,297)(278,300)
(279,299)(280,298)(281,293)(282,296)(283,295)(284,294)(285,289)(286,292)
(287,291)(288,290);
s3 := Sym(324)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)
(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)
(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)
(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)
(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)
(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)
(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)
(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)
(303,304)(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)
(319,320)(321,322)(323,324);
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,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope