Polytope of Type {8,9,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,9,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,30284)
Rank : 4
Schlafli Type : {8,9,6}
Number of vertices, edges, etc : 16, 72, 54, 6
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 : {4,9,6}*864
3-fold quotients : {8,9,2}*576, {8,3,6}*576
4-fold quotients : {4,9,6}*432
6-fold quotients : {4,9,2}*288, {4,3,6}*288
8-fold quotients : {2,9,6}*216
9-fold quotients : {8,3,2}*192
12-fold quotients : {4,9,2}*144, {4,3,6}*144
18-fold quotients : {4,3,2}*96
24-fold quotients : {2,9,2}*72, {2,3,6}*72
36-fold quotients : {4,3,2}*48
72-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,221)( 2,222)( 3,224)( 4,223)( 5,218)( 6,217)( 7,219)( 8,220)
( 9,229)( 10,230)( 11,232)( 12,231)( 13,226)( 14,225)( 15,227)( 16,228)
( 17,237)( 18,238)( 19,240)( 20,239)( 21,234)( 22,233)( 23,235)( 24,236)
( 25,245)( 26,246)( 27,248)( 28,247)( 29,242)( 30,241)( 31,243)( 32,244)
( 33,253)( 34,254)( 35,256)( 36,255)( 37,250)( 38,249)( 39,251)( 40,252)
( 41,261)( 42,262)( 43,264)( 44,263)( 45,258)( 46,257)( 47,259)( 48,260)
( 49,269)( 50,270)( 51,272)( 52,271)( 53,266)( 54,265)( 55,267)( 56,268)
( 57,277)( 58,278)( 59,280)( 60,279)( 61,274)( 62,273)( 63,275)( 64,276)
( 65,285)( 66,286)( 67,288)( 68,287)( 69,282)( 70,281)( 71,283)( 72,284)
( 73,293)( 74,294)( 75,296)( 76,295)( 77,290)( 78,289)( 79,291)( 80,292)
( 81,301)( 82,302)( 83,304)( 84,303)( 85,298)( 86,297)( 87,299)( 88,300)
( 89,309)( 90,310)( 91,312)( 92,311)( 93,306)( 94,305)( 95,307)( 96,308)
( 97,317)( 98,318)( 99,320)(100,319)(101,314)(102,313)(103,315)(104,316)
(105,325)(106,326)(107,328)(108,327)(109,322)(110,321)(111,323)(112,324)
(113,333)(114,334)(115,336)(116,335)(117,330)(118,329)(119,331)(120,332)
(121,341)(122,342)(123,344)(124,343)(125,338)(126,337)(127,339)(128,340)
(129,349)(130,350)(131,352)(132,351)(133,346)(134,345)(135,347)(136,348)
(137,357)(138,358)(139,360)(140,359)(141,354)(142,353)(143,355)(144,356)
(145,365)(146,366)(147,368)(148,367)(149,362)(150,361)(151,363)(152,364)
(153,373)(154,374)(155,376)(156,375)(157,370)(158,369)(159,371)(160,372)
(161,381)(162,382)(163,384)(164,383)(165,378)(166,377)(167,379)(168,380)
(169,389)(170,390)(171,392)(172,391)(173,386)(174,385)(175,387)(176,388)
(177,397)(178,398)(179,400)(180,399)(181,394)(182,393)(183,395)(184,396)
(185,405)(186,406)(187,408)(188,407)(189,402)(190,401)(191,403)(192,404)
(193,413)(194,414)(195,416)(196,415)(197,410)(198,409)(199,411)(200,412)
(201,421)(202,422)(203,424)(204,423)(205,418)(206,417)(207,419)(208,420)
(209,429)(210,430)(211,432)(212,431)(213,426)(214,425)(215,427)(216,428);;
s1 := ( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 13, 23)
( 14, 24)( 15, 21)( 16, 22)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 55)
( 30, 56)( 31, 53)( 32, 54)( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 37, 71)
( 38, 72)( 39, 69)( 40, 70)( 41, 57)( 42, 58)( 43, 60)( 44, 59)( 45, 63)
( 46, 64)( 47, 61)( 48, 62)( 73,161)( 74,162)( 75,164)( 76,163)( 77,167)
( 78,168)( 79,165)( 80,166)( 81,153)( 82,154)( 83,156)( 84,155)( 85,159)
( 86,160)( 87,157)( 88,158)( 89,145)( 90,146)( 91,148)( 92,147)( 93,151)
( 94,152)( 95,149)( 96,150)( 97,209)( 98,210)( 99,212)(100,211)(101,215)
(102,216)(103,213)(104,214)(105,201)(106,202)(107,204)(108,203)(109,207)
(110,208)(111,205)(112,206)(113,193)(114,194)(115,196)(116,195)(117,199)
(118,200)(119,197)(120,198)(121,185)(122,186)(123,188)(124,187)(125,191)
(126,192)(127,189)(128,190)(129,177)(130,178)(131,180)(132,179)(133,183)
(134,184)(135,181)(136,182)(137,169)(138,170)(139,172)(140,171)(141,175)
(142,176)(143,173)(144,174)(217,218)(221,224)(222,223)(225,234)(226,233)
(227,235)(228,236)(229,240)(230,239)(231,238)(232,237)(241,266)(242,265)
(243,267)(244,268)(245,272)(246,271)(247,270)(248,269)(249,282)(250,281)
(251,283)(252,284)(253,288)(254,287)(255,286)(256,285)(257,274)(258,273)
(259,275)(260,276)(261,280)(262,279)(263,278)(264,277)(289,378)(290,377)
(291,379)(292,380)(293,384)(294,383)(295,382)(296,381)(297,370)(298,369)
(299,371)(300,372)(301,376)(302,375)(303,374)(304,373)(305,362)(306,361)
(307,363)(308,364)(309,368)(310,367)(311,366)(312,365)(313,426)(314,425)
(315,427)(316,428)(317,432)(318,431)(319,430)(320,429)(321,418)(322,417)
(323,419)(324,420)(325,424)(326,423)(327,422)(328,421)(329,410)(330,409)
(331,411)(332,412)(333,416)(334,415)(335,414)(336,413)(337,402)(338,401)
(339,403)(340,404)(341,408)(342,407)(343,406)(344,405)(345,394)(346,393)
(347,395)(348,396)(349,400)(350,399)(351,398)(352,397)(353,386)(354,385)
(355,387)(356,388)(357,392)(358,391)(359,390)(360,389);;
s2 := ( 1, 97)( 2, 98)( 3,103)( 4,104)( 5,102)( 6,101)( 7, 99)( 8,100)
( 9,113)( 10,114)( 11,119)( 12,120)( 13,118)( 14,117)( 15,115)( 16,116)
( 17,105)( 18,106)( 19,111)( 20,112)( 21,110)( 22,109)( 23,107)( 24,108)
( 25, 73)( 26, 74)( 27, 79)( 28, 80)( 29, 78)( 30, 77)( 31, 75)( 32, 76)
( 33, 89)( 34, 90)( 35, 95)( 36, 96)( 37, 94)( 38, 93)( 39, 91)( 40, 92)
( 41, 81)( 42, 82)( 43, 87)( 44, 88)( 45, 86)( 46, 85)( 47, 83)( 48, 84)
( 49,121)( 50,122)( 51,127)( 52,128)( 53,126)( 54,125)( 55,123)( 56,124)
( 57,137)( 58,138)( 59,143)( 60,144)( 61,142)( 62,141)( 63,139)( 64,140)
( 65,129)( 66,130)( 67,135)( 68,136)( 69,134)( 70,133)( 71,131)( 72,132)
(145,185)(146,186)(147,191)(148,192)(149,190)(150,189)(151,187)(152,188)
(153,177)(154,178)(155,183)(156,184)(157,182)(158,181)(159,179)(160,180)
(161,169)(162,170)(163,175)(164,176)(165,174)(166,173)(167,171)(168,172)
(193,209)(194,210)(195,215)(196,216)(197,214)(198,213)(199,211)(200,212)
(203,207)(204,208)(205,206)(217,314)(218,313)(219,320)(220,319)(221,317)
(222,318)(223,316)(224,315)(225,330)(226,329)(227,336)(228,335)(229,333)
(230,334)(231,332)(232,331)(233,322)(234,321)(235,328)(236,327)(237,325)
(238,326)(239,324)(240,323)(241,290)(242,289)(243,296)(244,295)(245,293)
(246,294)(247,292)(248,291)(249,306)(250,305)(251,312)(252,311)(253,309)
(254,310)(255,308)(256,307)(257,298)(258,297)(259,304)(260,303)(261,301)
(262,302)(263,300)(264,299)(265,338)(266,337)(267,344)(268,343)(269,341)
(270,342)(271,340)(272,339)(273,354)(274,353)(275,360)(276,359)(277,357)
(278,358)(279,356)(280,355)(281,346)(282,345)(283,352)(284,351)(285,349)
(286,350)(287,348)(288,347)(361,402)(362,401)(363,408)(364,407)(365,405)
(366,406)(367,404)(368,403)(369,394)(370,393)(371,400)(372,399)(373,397)
(374,398)(375,396)(376,395)(377,386)(378,385)(379,392)(380,391)(381,389)
(382,390)(383,388)(384,387)(409,426)(410,425)(411,432)(412,431)(413,429)
(414,430)(415,428)(416,427)(417,418)(419,424)(420,423);;
s3 := ( 25, 49)( 26, 50)( 27, 51)( 28, 52)( 29, 53)( 30, 54)( 31, 55)( 32, 56)
( 33, 57)( 34, 58)( 35, 59)( 36, 60)( 37, 61)( 38, 62)( 39, 63)( 40, 64)
( 41, 65)( 42, 66)( 43, 67)( 44, 68)( 45, 69)( 46, 70)( 47, 71)( 48, 72)
( 97,121)( 98,122)( 99,123)(100,124)(101,125)(102,126)(103,127)(104,128)
(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)
(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144)
(169,193)(170,194)(171,195)(172,196)(173,197)(174,198)(175,199)(176,200)
(177,201)(178,202)(179,203)(180,204)(181,205)(182,206)(183,207)(184,208)
(185,209)(186,210)(187,211)(188,212)(189,213)(190,214)(191,215)(192,216)
(241,265)(242,266)(243,267)(244,268)(245,269)(246,270)(247,271)(248,272)
(249,273)(250,274)(251,275)(252,276)(253,277)(254,278)(255,279)(256,280)
(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)(264,288)
(313,337)(314,338)(315,339)(316,340)(317,341)(318,342)(319,343)(320,344)
(321,345)(322,346)(323,347)(324,348)(325,349)(326,350)(327,351)(328,352)
(329,353)(330,354)(331,355)(332,356)(333,357)(334,358)(335,359)(336,360)
(385,409)(386,410)(387,411)(388,412)(389,413)(390,414)(391,415)(392,416)
(393,417)(394,418)(395,419)(396,420)(397,421)(398,422)(399,423)(400,424)
(401,425)(402,426)(403,427)(404,428)(405,429)(406,430)(407,431)(408,432);;
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, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*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(432)!( 1,221)( 2,222)( 3,224)( 4,223)( 5,218)( 6,217)( 7,219)
( 8,220)( 9,229)( 10,230)( 11,232)( 12,231)( 13,226)( 14,225)( 15,227)
( 16,228)( 17,237)( 18,238)( 19,240)( 20,239)( 21,234)( 22,233)( 23,235)
( 24,236)( 25,245)( 26,246)( 27,248)( 28,247)( 29,242)( 30,241)( 31,243)
( 32,244)( 33,253)( 34,254)( 35,256)( 36,255)( 37,250)( 38,249)( 39,251)
( 40,252)( 41,261)( 42,262)( 43,264)( 44,263)( 45,258)( 46,257)( 47,259)
( 48,260)( 49,269)( 50,270)( 51,272)( 52,271)( 53,266)( 54,265)( 55,267)
( 56,268)( 57,277)( 58,278)( 59,280)( 60,279)( 61,274)( 62,273)( 63,275)
( 64,276)( 65,285)( 66,286)( 67,288)( 68,287)( 69,282)( 70,281)( 71,283)
( 72,284)( 73,293)( 74,294)( 75,296)( 76,295)( 77,290)( 78,289)( 79,291)
( 80,292)( 81,301)( 82,302)( 83,304)( 84,303)( 85,298)( 86,297)( 87,299)
( 88,300)( 89,309)( 90,310)( 91,312)( 92,311)( 93,306)( 94,305)( 95,307)
( 96,308)( 97,317)( 98,318)( 99,320)(100,319)(101,314)(102,313)(103,315)
(104,316)(105,325)(106,326)(107,328)(108,327)(109,322)(110,321)(111,323)
(112,324)(113,333)(114,334)(115,336)(116,335)(117,330)(118,329)(119,331)
(120,332)(121,341)(122,342)(123,344)(124,343)(125,338)(126,337)(127,339)
(128,340)(129,349)(130,350)(131,352)(132,351)(133,346)(134,345)(135,347)
(136,348)(137,357)(138,358)(139,360)(140,359)(141,354)(142,353)(143,355)
(144,356)(145,365)(146,366)(147,368)(148,367)(149,362)(150,361)(151,363)
(152,364)(153,373)(154,374)(155,376)(156,375)(157,370)(158,369)(159,371)
(160,372)(161,381)(162,382)(163,384)(164,383)(165,378)(166,377)(167,379)
(168,380)(169,389)(170,390)(171,392)(172,391)(173,386)(174,385)(175,387)
(176,388)(177,397)(178,398)(179,400)(180,399)(181,394)(182,393)(183,395)
(184,396)(185,405)(186,406)(187,408)(188,407)(189,402)(190,401)(191,403)
(192,404)(193,413)(194,414)(195,416)(196,415)(197,410)(198,409)(199,411)
(200,412)(201,421)(202,422)(203,424)(204,423)(205,418)(206,417)(207,419)
(208,420)(209,429)(210,430)(211,432)(212,431)(213,426)(214,425)(215,427)
(216,428);
s1 := Sym(432)!( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)
( 13, 23)( 14, 24)( 15, 21)( 16, 22)( 25, 49)( 26, 50)( 27, 52)( 28, 51)
( 29, 55)( 30, 56)( 31, 53)( 32, 54)( 33, 65)( 34, 66)( 35, 68)( 36, 67)
( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 57)( 42, 58)( 43, 60)( 44, 59)
( 45, 63)( 46, 64)( 47, 61)( 48, 62)( 73,161)( 74,162)( 75,164)( 76,163)
( 77,167)( 78,168)( 79,165)( 80,166)( 81,153)( 82,154)( 83,156)( 84,155)
( 85,159)( 86,160)( 87,157)( 88,158)( 89,145)( 90,146)( 91,148)( 92,147)
( 93,151)( 94,152)( 95,149)( 96,150)( 97,209)( 98,210)( 99,212)(100,211)
(101,215)(102,216)(103,213)(104,214)(105,201)(106,202)(107,204)(108,203)
(109,207)(110,208)(111,205)(112,206)(113,193)(114,194)(115,196)(116,195)
(117,199)(118,200)(119,197)(120,198)(121,185)(122,186)(123,188)(124,187)
(125,191)(126,192)(127,189)(128,190)(129,177)(130,178)(131,180)(132,179)
(133,183)(134,184)(135,181)(136,182)(137,169)(138,170)(139,172)(140,171)
(141,175)(142,176)(143,173)(144,174)(217,218)(221,224)(222,223)(225,234)
(226,233)(227,235)(228,236)(229,240)(230,239)(231,238)(232,237)(241,266)
(242,265)(243,267)(244,268)(245,272)(246,271)(247,270)(248,269)(249,282)
(250,281)(251,283)(252,284)(253,288)(254,287)(255,286)(256,285)(257,274)
(258,273)(259,275)(260,276)(261,280)(262,279)(263,278)(264,277)(289,378)
(290,377)(291,379)(292,380)(293,384)(294,383)(295,382)(296,381)(297,370)
(298,369)(299,371)(300,372)(301,376)(302,375)(303,374)(304,373)(305,362)
(306,361)(307,363)(308,364)(309,368)(310,367)(311,366)(312,365)(313,426)
(314,425)(315,427)(316,428)(317,432)(318,431)(319,430)(320,429)(321,418)
(322,417)(323,419)(324,420)(325,424)(326,423)(327,422)(328,421)(329,410)
(330,409)(331,411)(332,412)(333,416)(334,415)(335,414)(336,413)(337,402)
(338,401)(339,403)(340,404)(341,408)(342,407)(343,406)(344,405)(345,394)
(346,393)(347,395)(348,396)(349,400)(350,399)(351,398)(352,397)(353,386)
(354,385)(355,387)(356,388)(357,392)(358,391)(359,390)(360,389);
s2 := Sym(432)!( 1, 97)( 2, 98)( 3,103)( 4,104)( 5,102)( 6,101)( 7, 99)
( 8,100)( 9,113)( 10,114)( 11,119)( 12,120)( 13,118)( 14,117)( 15,115)
( 16,116)( 17,105)( 18,106)( 19,111)( 20,112)( 21,110)( 22,109)( 23,107)
( 24,108)( 25, 73)( 26, 74)( 27, 79)( 28, 80)( 29, 78)( 30, 77)( 31, 75)
( 32, 76)( 33, 89)( 34, 90)( 35, 95)( 36, 96)( 37, 94)( 38, 93)( 39, 91)
( 40, 92)( 41, 81)( 42, 82)( 43, 87)( 44, 88)( 45, 86)( 46, 85)( 47, 83)
( 48, 84)( 49,121)( 50,122)( 51,127)( 52,128)( 53,126)( 54,125)( 55,123)
( 56,124)( 57,137)( 58,138)( 59,143)( 60,144)( 61,142)( 62,141)( 63,139)
( 64,140)( 65,129)( 66,130)( 67,135)( 68,136)( 69,134)( 70,133)( 71,131)
( 72,132)(145,185)(146,186)(147,191)(148,192)(149,190)(150,189)(151,187)
(152,188)(153,177)(154,178)(155,183)(156,184)(157,182)(158,181)(159,179)
(160,180)(161,169)(162,170)(163,175)(164,176)(165,174)(166,173)(167,171)
(168,172)(193,209)(194,210)(195,215)(196,216)(197,214)(198,213)(199,211)
(200,212)(203,207)(204,208)(205,206)(217,314)(218,313)(219,320)(220,319)
(221,317)(222,318)(223,316)(224,315)(225,330)(226,329)(227,336)(228,335)
(229,333)(230,334)(231,332)(232,331)(233,322)(234,321)(235,328)(236,327)
(237,325)(238,326)(239,324)(240,323)(241,290)(242,289)(243,296)(244,295)
(245,293)(246,294)(247,292)(248,291)(249,306)(250,305)(251,312)(252,311)
(253,309)(254,310)(255,308)(256,307)(257,298)(258,297)(259,304)(260,303)
(261,301)(262,302)(263,300)(264,299)(265,338)(266,337)(267,344)(268,343)
(269,341)(270,342)(271,340)(272,339)(273,354)(274,353)(275,360)(276,359)
(277,357)(278,358)(279,356)(280,355)(281,346)(282,345)(283,352)(284,351)
(285,349)(286,350)(287,348)(288,347)(361,402)(362,401)(363,408)(364,407)
(365,405)(366,406)(367,404)(368,403)(369,394)(370,393)(371,400)(372,399)
(373,397)(374,398)(375,396)(376,395)(377,386)(378,385)(379,392)(380,391)
(381,389)(382,390)(383,388)(384,387)(409,426)(410,425)(411,432)(412,431)
(413,429)(414,430)(415,428)(416,427)(417,418)(419,424)(420,423);
s3 := Sym(432)!( 25, 49)( 26, 50)( 27, 51)( 28, 52)( 29, 53)( 30, 54)( 31, 55)
( 32, 56)( 33, 57)( 34, 58)( 35, 59)( 36, 60)( 37, 61)( 38, 62)( 39, 63)
( 40, 64)( 41, 65)( 42, 66)( 43, 67)( 44, 68)( 45, 69)( 46, 70)( 47, 71)
( 48, 72)( 97,121)( 98,122)( 99,123)(100,124)(101,125)(102,126)(103,127)
(104,128)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)
(112,136)(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)
(120,144)(169,193)(170,194)(171,195)(172,196)(173,197)(174,198)(175,199)
(176,200)(177,201)(178,202)(179,203)(180,204)(181,205)(182,206)(183,207)
(184,208)(185,209)(186,210)(187,211)(188,212)(189,213)(190,214)(191,215)
(192,216)(241,265)(242,266)(243,267)(244,268)(245,269)(246,270)(247,271)
(248,272)(249,273)(250,274)(251,275)(252,276)(253,277)(254,278)(255,279)
(256,280)(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)
(264,288)(313,337)(314,338)(315,339)(316,340)(317,341)(318,342)(319,343)
(320,344)(321,345)(322,346)(323,347)(324,348)(325,349)(326,350)(327,351)
(328,352)(329,353)(330,354)(331,355)(332,356)(333,357)(334,358)(335,359)
(336,360)(385,409)(386,410)(387,411)(388,412)(389,413)(390,414)(391,415)
(392,416)(393,417)(394,418)(395,419)(396,420)(397,421)(398,422)(399,423)
(400,424)(401,425)(402,426)(403,427)(404,428)(405,429)(406,430)(407,431)
(408,432);
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,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*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