Polytope of Type {3,12,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,12,18}*1728
if this polytope has a name.
Group : SmallGroup(1728,30201)
Rank : 4
Schlafli Type : {3,12,18}
Number of vertices, edges, etc : 4, 24, 144, 18
Order of s0s1s2s3 : 72
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 : {3,6,18}*864
3-fold quotients : {3,12,6}*576
6-fold quotients : {3,6,6}*288
9-fold quotients : {3,12,2}*192
18-fold quotients : {3,6,2}*96
36-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 7)( 6, 8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)( 21, 23)
( 22, 24)( 27, 28)( 29, 31)( 30, 32)( 35, 36)( 37, 39)( 38, 40)( 43, 44)
( 45, 47)( 46, 48)( 51, 52)( 53, 55)( 54, 56)( 59, 60)( 61, 63)( 62, 64)
( 67, 68)( 69, 71)( 70, 72)( 73,145)( 74,146)( 75,148)( 76,147)( 77,151)
( 78,152)( 79,149)( 80,150)( 81,153)( 82,154)( 83,156)( 84,155)( 85,159)
( 86,160)( 87,157)( 88,158)( 89,161)( 90,162)( 91,164)( 92,163)( 93,167)
( 94,168)( 95,165)( 96,166)( 97,169)( 98,170)( 99,172)(100,171)(101,175)
(102,176)(103,173)(104,174)(105,177)(106,178)(107,180)(108,179)(109,183)
(110,184)(111,181)(112,182)(113,185)(114,186)(115,188)(116,187)(117,191)
(118,192)(119,189)(120,190)(121,193)(122,194)(123,196)(124,195)(125,199)
(126,200)(127,197)(128,198)(129,201)(130,202)(131,204)(132,203)(133,207)
(134,208)(135,205)(136,206)(137,209)(138,210)(139,212)(140,211)(141,215)
(142,216)(143,213)(144,214)(217,218)(221,224)(222,223)(225,226)(229,232)
(230,231)(233,234)(237,240)(238,239)(241,242)(245,248)(246,247)(249,250)
(253,256)(254,255)(257,258)(261,264)(262,263)(265,266)(269,272)(270,271)
(273,274)(277,280)(278,279)(281,282)(285,288)(286,287)(289,362)(290,361)
(291,363)(292,364)(293,368)(294,367)(295,366)(296,365)(297,370)(298,369)
(299,371)(300,372)(301,376)(302,375)(303,374)(304,373)(305,378)(306,377)
(307,379)(308,380)(309,384)(310,383)(311,382)(312,381)(313,386)(314,385)
(315,387)(316,388)(317,392)(318,391)(319,390)(320,389)(321,394)(322,393)
(323,395)(324,396)(325,400)(326,399)(327,398)(328,397)(329,402)(330,401)
(331,403)(332,404)(333,408)(334,407)(335,406)(336,405)(337,410)(338,409)
(339,411)(340,412)(341,416)(342,415)(343,414)(344,413)(345,418)(346,417)
(347,419)(348,420)(349,424)(350,423)(351,422)(352,421)(353,426)(354,425)
(355,427)(356,428)(357,432)(358,431)(359,430)(360,429);;
s1 := ( 1, 73)( 2, 74)( 3, 79)( 4, 80)( 5, 78)( 6, 77)( 7, 75)( 8, 76)
( 9, 81)( 10, 82)( 11, 87)( 12, 88)( 13, 86)( 14, 85)( 15, 83)( 16, 84)
( 17, 89)( 18, 90)( 19, 95)( 20, 96)( 21, 94)( 22, 93)( 23, 91)( 24, 92)
( 25, 97)( 26, 98)( 27,103)( 28,104)( 29,102)( 30,101)( 31, 99)( 32,100)
( 33,105)( 34,106)( 35,111)( 36,112)( 37,110)( 38,109)( 39,107)( 40,108)
( 41,113)( 42,114)( 43,119)( 44,120)( 45,118)( 46,117)( 47,115)( 48,116)
( 49,121)( 50,122)( 51,127)( 52,128)( 53,126)( 54,125)( 55,123)( 56,124)
( 57,129)( 58,130)( 59,135)( 60,136)( 61,134)( 62,133)( 63,131)( 64,132)
( 65,137)( 66,138)( 67,143)( 68,144)( 69,142)( 70,141)( 71,139)( 72,140)
(147,151)(148,152)(149,150)(155,159)(156,160)(157,158)(163,167)(164,168)
(165,166)(171,175)(172,176)(173,174)(179,183)(180,184)(181,182)(187,191)
(188,192)(189,190)(195,199)(196,200)(197,198)(203,207)(204,208)(205,206)
(211,215)(212,216)(213,214)(217,290)(218,289)(219,296)(220,295)(221,293)
(222,294)(223,292)(224,291)(225,298)(226,297)(227,304)(228,303)(229,301)
(230,302)(231,300)(232,299)(233,306)(234,305)(235,312)(236,311)(237,309)
(238,310)(239,308)(240,307)(241,314)(242,313)(243,320)(244,319)(245,317)
(246,318)(247,316)(248,315)(249,322)(250,321)(251,328)(252,327)(253,325)
(254,326)(255,324)(256,323)(257,330)(258,329)(259,336)(260,335)(261,333)
(262,334)(263,332)(264,331)(265,338)(266,337)(267,344)(268,343)(269,341)
(270,342)(271,340)(272,339)(273,346)(274,345)(275,352)(276,351)(277,349)
(278,350)(279,348)(280,347)(281,354)(282,353)(283,360)(284,359)(285,357)
(286,358)(287,356)(288,355)(361,362)(363,368)(364,367)(369,370)(371,376)
(372,375)(377,378)(379,384)(380,383)(385,386)(387,392)(388,391)(393,394)
(395,400)(396,399)(401,402)(403,408)(404,407)(409,410)(411,416)(412,415)
(417,418)(419,424)(420,423)(425,426)(427,432)(428,431);;
s2 := ( 1,219)( 2,220)( 3,217)( 4,218)( 5,222)( 6,221)( 7,223)( 8,224)
( 9,235)( 10,236)( 11,233)( 12,234)( 13,238)( 14,237)( 15,239)( 16,240)
( 17,227)( 18,228)( 19,225)( 20,226)( 21,230)( 22,229)( 23,231)( 24,232)
( 25,283)( 26,284)( 27,281)( 28,282)( 29,286)( 30,285)( 31,287)( 32,288)
( 33,275)( 34,276)( 35,273)( 36,274)( 37,278)( 38,277)( 39,279)( 40,280)
( 41,267)( 42,268)( 43,265)( 44,266)( 45,270)( 46,269)( 47,271)( 48,272)
( 49,259)( 50,260)( 51,257)( 52,258)( 53,262)( 54,261)( 55,263)( 56,264)
( 57,251)( 58,252)( 59,249)( 60,250)( 61,254)( 62,253)( 63,255)( 64,256)
( 65,243)( 66,244)( 67,241)( 68,242)( 69,246)( 70,245)( 71,247)( 72,248)
( 73,363)( 74,364)( 75,361)( 76,362)( 77,366)( 78,365)( 79,367)( 80,368)
( 81,379)( 82,380)( 83,377)( 84,378)( 85,382)( 86,381)( 87,383)( 88,384)
( 89,371)( 90,372)( 91,369)( 92,370)( 93,374)( 94,373)( 95,375)( 96,376)
( 97,427)( 98,428)( 99,425)(100,426)(101,430)(102,429)(103,431)(104,432)
(105,419)(106,420)(107,417)(108,418)(109,422)(110,421)(111,423)(112,424)
(113,411)(114,412)(115,409)(116,410)(117,414)(118,413)(119,415)(120,416)
(121,403)(122,404)(123,401)(124,402)(125,406)(126,405)(127,407)(128,408)
(129,395)(130,396)(131,393)(132,394)(133,398)(134,397)(135,399)(136,400)
(137,387)(138,388)(139,385)(140,386)(141,390)(142,389)(143,391)(144,392)
(145,291)(146,292)(147,289)(148,290)(149,294)(150,293)(151,295)(152,296)
(153,307)(154,308)(155,305)(156,306)(157,310)(158,309)(159,311)(160,312)
(161,299)(162,300)(163,297)(164,298)(165,302)(166,301)(167,303)(168,304)
(169,355)(170,356)(171,353)(172,354)(173,358)(174,357)(175,359)(176,360)
(177,347)(178,348)(179,345)(180,346)(181,350)(182,349)(183,351)(184,352)
(185,339)(186,340)(187,337)(188,338)(189,342)(190,341)(191,343)(192,344)
(193,331)(194,332)(195,329)(196,330)(197,334)(198,333)(199,335)(200,336)
(201,323)(202,324)(203,321)(204,322)(205,326)(206,325)(207,327)(208,328)
(209,315)(210,316)(211,313)(212,314)(213,318)(214,317)(215,319)(216,320);;
s3 := ( 1, 25)( 2, 26)( 3, 27)( 4, 28)( 5, 29)( 6, 30)( 7, 31)( 8, 32)
( 9, 41)( 10, 42)( 11, 43)( 12, 44)( 13, 45)( 14, 46)( 15, 47)( 16, 48)
( 17, 33)( 18, 34)( 19, 35)( 20, 36)( 21, 37)( 22, 38)( 23, 39)( 24, 40)
( 49, 65)( 50, 66)( 51, 67)( 52, 68)( 53, 69)( 54, 70)( 55, 71)( 56, 72)
( 73, 97)( 74, 98)( 75, 99)( 76,100)( 77,101)( 78,102)( 79,103)( 80,104)
( 81,113)( 82,114)( 83,115)( 84,116)( 85,117)( 86,118)( 87,119)( 88,120)
( 89,105)( 90,106)( 91,107)( 92,108)( 93,109)( 94,110)( 95,111)( 96,112)
(121,137)(122,138)(123,139)(124,140)(125,141)(126,142)(127,143)(128,144)
(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)
(153,185)(154,186)(155,187)(156,188)(157,189)(158,190)(159,191)(160,192)
(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,183)(168,184)
(193,209)(194,210)(195,211)(196,212)(197,213)(198,214)(199,215)(200,216)
(217,241)(218,242)(219,243)(220,244)(221,245)(222,246)(223,247)(224,248)
(225,257)(226,258)(227,259)(228,260)(229,261)(230,262)(231,263)(232,264)
(233,249)(234,250)(235,251)(236,252)(237,253)(238,254)(239,255)(240,256)
(265,281)(266,282)(267,283)(268,284)(269,285)(270,286)(271,287)(272,288)
(289,313)(290,314)(291,315)(292,316)(293,317)(294,318)(295,319)(296,320)
(297,329)(298,330)(299,331)(300,332)(301,333)(302,334)(303,335)(304,336)
(305,321)(306,322)(307,323)(308,324)(309,325)(310,326)(311,327)(312,328)
(337,353)(338,354)(339,355)(340,356)(341,357)(342,358)(343,359)(344,360)
(361,385)(362,386)(363,387)(364,388)(365,389)(366,390)(367,391)(368,392)
(369,401)(370,402)(371,403)(372,404)(373,405)(374,406)(375,407)(376,408)
(377,393)(378,394)(379,395)(380,396)(381,397)(382,398)(383,399)(384,400)
(409,425)(410,426)(411,427)(412,428)(413,429)(414,430)(415,431)(416,432);;
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, s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*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(432)!( 3, 4)( 5, 7)( 6, 8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)
( 21, 23)( 22, 24)( 27, 28)( 29, 31)( 30, 32)( 35, 36)( 37, 39)( 38, 40)
( 43, 44)( 45, 47)( 46, 48)( 51, 52)( 53, 55)( 54, 56)( 59, 60)( 61, 63)
( 62, 64)( 67, 68)( 69, 71)( 70, 72)( 73,145)( 74,146)( 75,148)( 76,147)
( 77,151)( 78,152)( 79,149)( 80,150)( 81,153)( 82,154)( 83,156)( 84,155)
( 85,159)( 86,160)( 87,157)( 88,158)( 89,161)( 90,162)( 91,164)( 92,163)
( 93,167)( 94,168)( 95,165)( 96,166)( 97,169)( 98,170)( 99,172)(100,171)
(101,175)(102,176)(103,173)(104,174)(105,177)(106,178)(107,180)(108,179)
(109,183)(110,184)(111,181)(112,182)(113,185)(114,186)(115,188)(116,187)
(117,191)(118,192)(119,189)(120,190)(121,193)(122,194)(123,196)(124,195)
(125,199)(126,200)(127,197)(128,198)(129,201)(130,202)(131,204)(132,203)
(133,207)(134,208)(135,205)(136,206)(137,209)(138,210)(139,212)(140,211)
(141,215)(142,216)(143,213)(144,214)(217,218)(221,224)(222,223)(225,226)
(229,232)(230,231)(233,234)(237,240)(238,239)(241,242)(245,248)(246,247)
(249,250)(253,256)(254,255)(257,258)(261,264)(262,263)(265,266)(269,272)
(270,271)(273,274)(277,280)(278,279)(281,282)(285,288)(286,287)(289,362)
(290,361)(291,363)(292,364)(293,368)(294,367)(295,366)(296,365)(297,370)
(298,369)(299,371)(300,372)(301,376)(302,375)(303,374)(304,373)(305,378)
(306,377)(307,379)(308,380)(309,384)(310,383)(311,382)(312,381)(313,386)
(314,385)(315,387)(316,388)(317,392)(318,391)(319,390)(320,389)(321,394)
(322,393)(323,395)(324,396)(325,400)(326,399)(327,398)(328,397)(329,402)
(330,401)(331,403)(332,404)(333,408)(334,407)(335,406)(336,405)(337,410)
(338,409)(339,411)(340,412)(341,416)(342,415)(343,414)(344,413)(345,418)
(346,417)(347,419)(348,420)(349,424)(350,423)(351,422)(352,421)(353,426)
(354,425)(355,427)(356,428)(357,432)(358,431)(359,430)(360,429);
s1 := Sym(432)!( 1, 73)( 2, 74)( 3, 79)( 4, 80)( 5, 78)( 6, 77)( 7, 75)
( 8, 76)( 9, 81)( 10, 82)( 11, 87)( 12, 88)( 13, 86)( 14, 85)( 15, 83)
( 16, 84)( 17, 89)( 18, 90)( 19, 95)( 20, 96)( 21, 94)( 22, 93)( 23, 91)
( 24, 92)( 25, 97)( 26, 98)( 27,103)( 28,104)( 29,102)( 30,101)( 31, 99)
( 32,100)( 33,105)( 34,106)( 35,111)( 36,112)( 37,110)( 38,109)( 39,107)
( 40,108)( 41,113)( 42,114)( 43,119)( 44,120)( 45,118)( 46,117)( 47,115)
( 48,116)( 49,121)( 50,122)( 51,127)( 52,128)( 53,126)( 54,125)( 55,123)
( 56,124)( 57,129)( 58,130)( 59,135)( 60,136)( 61,134)( 62,133)( 63,131)
( 64,132)( 65,137)( 66,138)( 67,143)( 68,144)( 69,142)( 70,141)( 71,139)
( 72,140)(147,151)(148,152)(149,150)(155,159)(156,160)(157,158)(163,167)
(164,168)(165,166)(171,175)(172,176)(173,174)(179,183)(180,184)(181,182)
(187,191)(188,192)(189,190)(195,199)(196,200)(197,198)(203,207)(204,208)
(205,206)(211,215)(212,216)(213,214)(217,290)(218,289)(219,296)(220,295)
(221,293)(222,294)(223,292)(224,291)(225,298)(226,297)(227,304)(228,303)
(229,301)(230,302)(231,300)(232,299)(233,306)(234,305)(235,312)(236,311)
(237,309)(238,310)(239,308)(240,307)(241,314)(242,313)(243,320)(244,319)
(245,317)(246,318)(247,316)(248,315)(249,322)(250,321)(251,328)(252,327)
(253,325)(254,326)(255,324)(256,323)(257,330)(258,329)(259,336)(260,335)
(261,333)(262,334)(263,332)(264,331)(265,338)(266,337)(267,344)(268,343)
(269,341)(270,342)(271,340)(272,339)(273,346)(274,345)(275,352)(276,351)
(277,349)(278,350)(279,348)(280,347)(281,354)(282,353)(283,360)(284,359)
(285,357)(286,358)(287,356)(288,355)(361,362)(363,368)(364,367)(369,370)
(371,376)(372,375)(377,378)(379,384)(380,383)(385,386)(387,392)(388,391)
(393,394)(395,400)(396,399)(401,402)(403,408)(404,407)(409,410)(411,416)
(412,415)(417,418)(419,424)(420,423)(425,426)(427,432)(428,431);
s2 := Sym(432)!( 1,219)( 2,220)( 3,217)( 4,218)( 5,222)( 6,221)( 7,223)
( 8,224)( 9,235)( 10,236)( 11,233)( 12,234)( 13,238)( 14,237)( 15,239)
( 16,240)( 17,227)( 18,228)( 19,225)( 20,226)( 21,230)( 22,229)( 23,231)
( 24,232)( 25,283)( 26,284)( 27,281)( 28,282)( 29,286)( 30,285)( 31,287)
( 32,288)( 33,275)( 34,276)( 35,273)( 36,274)( 37,278)( 38,277)( 39,279)
( 40,280)( 41,267)( 42,268)( 43,265)( 44,266)( 45,270)( 46,269)( 47,271)
( 48,272)( 49,259)( 50,260)( 51,257)( 52,258)( 53,262)( 54,261)( 55,263)
( 56,264)( 57,251)( 58,252)( 59,249)( 60,250)( 61,254)( 62,253)( 63,255)
( 64,256)( 65,243)( 66,244)( 67,241)( 68,242)( 69,246)( 70,245)( 71,247)
( 72,248)( 73,363)( 74,364)( 75,361)( 76,362)( 77,366)( 78,365)( 79,367)
( 80,368)( 81,379)( 82,380)( 83,377)( 84,378)( 85,382)( 86,381)( 87,383)
( 88,384)( 89,371)( 90,372)( 91,369)( 92,370)( 93,374)( 94,373)( 95,375)
( 96,376)( 97,427)( 98,428)( 99,425)(100,426)(101,430)(102,429)(103,431)
(104,432)(105,419)(106,420)(107,417)(108,418)(109,422)(110,421)(111,423)
(112,424)(113,411)(114,412)(115,409)(116,410)(117,414)(118,413)(119,415)
(120,416)(121,403)(122,404)(123,401)(124,402)(125,406)(126,405)(127,407)
(128,408)(129,395)(130,396)(131,393)(132,394)(133,398)(134,397)(135,399)
(136,400)(137,387)(138,388)(139,385)(140,386)(141,390)(142,389)(143,391)
(144,392)(145,291)(146,292)(147,289)(148,290)(149,294)(150,293)(151,295)
(152,296)(153,307)(154,308)(155,305)(156,306)(157,310)(158,309)(159,311)
(160,312)(161,299)(162,300)(163,297)(164,298)(165,302)(166,301)(167,303)
(168,304)(169,355)(170,356)(171,353)(172,354)(173,358)(174,357)(175,359)
(176,360)(177,347)(178,348)(179,345)(180,346)(181,350)(182,349)(183,351)
(184,352)(185,339)(186,340)(187,337)(188,338)(189,342)(190,341)(191,343)
(192,344)(193,331)(194,332)(195,329)(196,330)(197,334)(198,333)(199,335)
(200,336)(201,323)(202,324)(203,321)(204,322)(205,326)(206,325)(207,327)
(208,328)(209,315)(210,316)(211,313)(212,314)(213,318)(214,317)(215,319)
(216,320);
s3 := Sym(432)!( 1, 25)( 2, 26)( 3, 27)( 4, 28)( 5, 29)( 6, 30)( 7, 31)
( 8, 32)( 9, 41)( 10, 42)( 11, 43)( 12, 44)( 13, 45)( 14, 46)( 15, 47)
( 16, 48)( 17, 33)( 18, 34)( 19, 35)( 20, 36)( 21, 37)( 22, 38)( 23, 39)
( 24, 40)( 49, 65)( 50, 66)( 51, 67)( 52, 68)( 53, 69)( 54, 70)( 55, 71)
( 56, 72)( 73, 97)( 74, 98)( 75, 99)( 76,100)( 77,101)( 78,102)( 79,103)
( 80,104)( 81,113)( 82,114)( 83,115)( 84,116)( 85,117)( 86,118)( 87,119)
( 88,120)( 89,105)( 90,106)( 91,107)( 92,108)( 93,109)( 94,110)( 95,111)
( 96,112)(121,137)(122,138)(123,139)(124,140)(125,141)(126,142)(127,143)
(128,144)(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)(151,175)
(152,176)(153,185)(154,186)(155,187)(156,188)(157,189)(158,190)(159,191)
(160,192)(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,183)
(168,184)(193,209)(194,210)(195,211)(196,212)(197,213)(198,214)(199,215)
(200,216)(217,241)(218,242)(219,243)(220,244)(221,245)(222,246)(223,247)
(224,248)(225,257)(226,258)(227,259)(228,260)(229,261)(230,262)(231,263)
(232,264)(233,249)(234,250)(235,251)(236,252)(237,253)(238,254)(239,255)
(240,256)(265,281)(266,282)(267,283)(268,284)(269,285)(270,286)(271,287)
(272,288)(289,313)(290,314)(291,315)(292,316)(293,317)(294,318)(295,319)
(296,320)(297,329)(298,330)(299,331)(300,332)(301,333)(302,334)(303,335)
(304,336)(305,321)(306,322)(307,323)(308,324)(309,325)(310,326)(311,327)
(312,328)(337,353)(338,354)(339,355)(340,356)(341,357)(342,358)(343,359)
(344,360)(361,385)(362,386)(363,387)(364,388)(365,389)(366,390)(367,391)
(368,392)(369,401)(370,402)(371,403)(372,404)(373,405)(374,406)(375,407)
(376,408)(377,393)(378,394)(379,395)(380,396)(381,397)(382,398)(383,399)
(384,400)(409,425)(410,426)(411,427)(412,428)(413,429)(414,430)(415,431)
(416,432);
poly := sub<Sym(432)|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,
s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*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