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