Polytope of Type {6,24,2}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,24,2}*1728e
if this polytope has a name.
Group : SmallGroup(1728,30374)
Rank : 4
Schlafli Type : {6,24,2}
Number of vertices, edges, etc : 18, 216, 72, 2
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
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 : {6,12,2}*864f
3-fold quotients : {6,8,2}*576
4-fold quotients : {6,12,2}*432b
6-fold quotients : {6,4,2}*288
12-fold quotients : {6,4,2}*144
27-fold quotients : {2,8,2}*64
54-fold quotients : {2,4,2}*32
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,163)( 2,164)( 3,165)( 4,171)( 5,169)( 6,170)( 7,167)( 8,168)( 9,166)( 10,181)( 11,182)( 12,183)( 13,189)( 14,187)( 15,188)( 16,185)( 17,186)( 18,184)( 19,172)( 20,173)( 21,174)( 22,180)( 23,178)( 24,179)( 25,176)( 26,177)( 27,175)( 28,190)( 29,191)( 30,192)( 31,198)( 32,196)( 33,197)( 34,194)( 35,195)( 36,193)( 37,208)( 38,209)( 39,210)( 40,216)( 41,214)( 42,215)( 43,212)( 44,213)( 45,211)( 46,199)( 47,200)( 48,201)( 49,207)( 50,205)( 51,206)( 52,203)( 53,204)( 54,202)( 55,136)( 56,137)( 57,138)( 58,144)( 59,142)( 60,143)( 61,140)( 62,141)( 63,139)( 64,154)( 65,155)( 66,156)( 67,162)( 68,160)( 69,161)( 70,158)( 71,159)( 72,157)( 73,145)( 74,146)( 75,147)( 76,153)( 77,151)( 78,152)( 79,149)( 80,150)( 81,148)( 82,109)( 83,110)( 84,111)( 85,117)( 86,115)( 87,116)( 88,113)( 89,114)( 90,112)( 91,127)( 92,128)( 93,129)( 94,135)( 95,133)( 96,134)( 97,131)( 98,132)( 99,130)(100,118)(101,119)(102,120)(103,126)(104,124)(105,125)(106,122)(107,123)(108,121)(217,379)(218,380)(219,381)(220,387)(221,385)(222,386)(223,383)(224,384)(225,382)(226,397)(227,398)(228,399)(229,405)(230,403)(231,404)(232,401)(233,402)(234,400)(235,388)(236,389)(237,390)(238,396)(239,394)(240,395)(241,392)(242,393)(243,391)(244,406)(245,407)(246,408)(247,414)(248,412)(249,413)(250,410)(251,411)(252,409)(253,424)(254,425)(255,426)(256,432)(257,430)(258,431)(259,428)(260,429)(261,427)(262,415)(263,416)(264,417)(265,423)(266,421)(267,422)(268,419)(269,420)(270,418)(271,352)(272,353)(273,354)(274,360)(275,358)(276,359)(277,356)(278,357)(279,355)(280,370)(281,371)(282,372)(283,378)(284,376)(285,377)(286,374)(287,375)(288,373)(289,361)(290,362)(291,363)(292,369)(293,367)(294,368)(295,365)(296,366)(297,364)(298,325)(299,326)(300,327)(301,333)(302,331)(303,332)(304,329)(305,330)(306,328)(307,343)(308,344)(309,345)(310,351)(311,349)(312,350)(313,347)(314,348)(315,346)(316,334)(317,335)(318,336)(319,342)(320,340)(321,341)(322,338)(323,339)(324,337);;
s1 := ( 1, 10)( 2, 12)( 3, 11)( 4, 15)( 5, 14)( 6, 13)( 7, 17)( 8, 16)( 9, 18)( 20, 21)( 22, 24)( 25, 26)( 28, 37)( 29, 39)( 30, 38)( 31, 42)( 32, 41)( 33, 40)( 34, 44)( 35, 43)( 36, 45)( 47, 48)( 49, 51)( 52, 53)( 55, 91)( 56, 93)( 57, 92)( 58, 96)( 59, 95)( 60, 94)( 61, 98)( 62, 97)( 63, 99)( 64, 82)( 65, 84)( 66, 83)( 67, 87)( 68, 86)( 69, 85)( 70, 89)( 71, 88)( 72, 90)( 73,100)( 74,102)( 75,101)( 76,105)( 77,104)( 78,103)( 79,107)( 80,106)( 81,108)(109,145)(110,147)(111,146)(112,150)(113,149)(114,148)(115,152)(116,151)(117,153)(118,136)(119,138)(120,137)(121,141)(122,140)(123,139)(124,143)(125,142)(126,144)(127,154)(128,156)(129,155)(130,159)(131,158)(132,157)(133,161)(134,160)(135,162)(163,172)(164,174)(165,173)(166,177)(167,176)(168,175)(169,179)(170,178)(171,180)(182,183)(184,186)(187,188)(190,199)(191,201)(192,200)(193,204)(194,203)(195,202)(196,206)(197,205)(198,207)(209,210)(211,213)(214,215)(217,280)(218,282)(219,281)(220,285)(221,284)(222,283)(223,287)(224,286)(225,288)(226,271)(227,273)(228,272)(229,276)(230,275)(231,274)(232,278)(233,277)(234,279)(235,289)(236,291)(237,290)(238,294)(239,293)(240,292)(241,296)(242,295)(243,297)(244,307)(245,309)(246,308)(247,312)(248,311)(249,310)(250,314)(251,313)(252,315)(253,298)(254,300)(255,299)(256,303)(257,302)(258,301)(259,305)(260,304)(261,306)(262,316)(263,318)(264,317)(265,321)(266,320)(267,319)(268,323)(269,322)(270,324)(325,415)(326,417)(327,416)(328,420)(329,419)(330,418)(331,422)(332,421)(333,423)(334,406)(335,408)(336,407)(337,411)(338,410)(339,409)(340,413)(341,412)(342,414)(343,424)(344,426)(345,425)(346,429)(347,428)(348,427)(349,431)(350,430)(351,432)(352,388)(353,390)(354,389)(355,393)(356,392)(357,391)(358,395)(359,394)(360,396)(361,379)(362,381)(363,380)(364,384)(365,383)(366,382)(367,386)(368,385)(369,387)(370,397)(371,399)(372,398)(373,402)(374,401)(375,400)(376,404)(377,403)(378,405);;
s2 := ( 1,272)( 2,271)( 3,273)( 4,289)( 5,291)( 6,290)( 7,282)( 8,281)( 9,280)( 10,279)( 11,278)( 12,277)( 13,297)( 14,296)( 15,295)( 16,288)( 17,287)( 18,286)( 19,274)( 20,276)( 21,275)( 22,293)( 23,292)( 24,294)( 25,285)( 26,284)( 27,283)( 28,299)( 29,298)( 30,300)( 31,316)( 32,318)( 33,317)( 34,309)( 35,308)( 36,307)( 37,306)( 38,305)( 39,304)( 40,324)( 41,323)( 42,322)( 43,315)( 44,314)( 45,313)( 46,301)( 47,303)( 48,302)( 49,320)( 50,319)( 51,321)( 52,312)( 53,311)( 54,310)( 55,218)( 56,217)( 57,219)( 58,235)( 59,237)( 60,236)( 61,228)( 62,227)( 63,226)( 64,225)( 65,224)( 66,223)( 67,243)( 68,242)( 69,241)( 70,234)( 71,233)( 72,232)( 73,220)( 74,222)( 75,221)( 76,239)( 77,238)( 78,240)( 79,231)( 80,230)( 81,229)( 82,245)( 83,244)( 84,246)( 85,262)( 86,264)( 87,263)( 88,255)( 89,254)( 90,253)( 91,252)( 92,251)( 93,250)( 94,270)( 95,269)( 96,268)( 97,261)( 98,260)( 99,259)(100,247)(101,249)(102,248)(103,266)(104,265)(105,267)(106,258)(107,257)(108,256)(109,407)(110,406)(111,408)(112,424)(113,426)(114,425)(115,417)(116,416)(117,415)(118,414)(119,413)(120,412)(121,432)(122,431)(123,430)(124,423)(125,422)(126,421)(127,409)(128,411)(129,410)(130,428)(131,427)(132,429)(133,420)(134,419)(135,418)(136,380)(137,379)(138,381)(139,397)(140,399)(141,398)(142,390)(143,389)(144,388)(145,387)(146,386)(147,385)(148,405)(149,404)(150,403)(151,396)(152,395)(153,394)(154,382)(155,384)(156,383)(157,401)(158,400)(159,402)(160,393)(161,392)(162,391)(163,353)(164,352)(165,354)(166,370)(167,372)(168,371)(169,363)(170,362)(171,361)(172,360)(173,359)(174,358)(175,378)(176,377)(177,376)(178,369)(179,368)(180,367)(181,355)(182,357)(183,356)(184,374)(185,373)(186,375)(187,366)(188,365)(189,364)(190,326)(191,325)(192,327)(193,343)(194,345)(195,344)(196,336)(197,335)(198,334)(199,333)(200,332)(201,331)(202,351)(203,350)(204,349)(205,342)(206,341)(207,340)(208,328)(209,330)(210,329)(211,347)(212,346)(213,348)(214,339)(215,338)(216,337);;
s3 := (433,434);;
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(434)!( 1,163)( 2,164)( 3,165)( 4,171)( 5,169)( 6,170)( 7,167)( 8,168)( 9,166)( 10,181)( 11,182)( 12,183)( 13,189)( 14,187)( 15,188)( 16,185)( 17,186)( 18,184)( 19,172)( 20,173)( 21,174)( 22,180)( 23,178)( 24,179)( 25,176)( 26,177)( 27,175)( 28,190)( 29,191)( 30,192)( 31,198)( 32,196)( 33,197)( 34,194)( 35,195)( 36,193)( 37,208)( 38,209)( 39,210)( 40,216)( 41,214)( 42,215)( 43,212)( 44,213)( 45,211)( 46,199)( 47,200)( 48,201)( 49,207)( 50,205)( 51,206)( 52,203)( 53,204)( 54,202)( 55,136)( 56,137)( 57,138)( 58,144)( 59,142)( 60,143)( 61,140)( 62,141)( 63,139)( 64,154)( 65,155)( 66,156)( 67,162)( 68,160)( 69,161)( 70,158)( 71,159)( 72,157)( 73,145)( 74,146)( 75,147)( 76,153)( 77,151)( 78,152)( 79,149)( 80,150)( 81,148)( 82,109)( 83,110)( 84,111)( 85,117)( 86,115)( 87,116)( 88,113)( 89,114)( 90,112)( 91,127)( 92,128)( 93,129)( 94,135)( 95,133)( 96,134)( 97,131)( 98,132)( 99,130)(100,118)(101,119)(102,120)(103,126)(104,124)(105,125)(106,122)(107,123)(108,121)(217,379)(218,380)(219,381)(220,387)(221,385)(222,386)(223,383)(224,384)(225,382)(226,397)(227,398)(228,399)(229,405)(230,403)(231,404)(232,401)(233,402)(234,400)(235,388)(236,389)(237,390)(238,396)(239,394)(240,395)(241,392)(242,393)(243,391)(244,406)(245,407)(246,408)(247,414)(248,412)(249,413)(250,410)(251,411)(252,409)(253,424)(254,425)(255,426)(256,432)(257,430)(258,431)(259,428)(260,429)(261,427)(262,415)(263,416)(264,417)(265,423)(266,421)(267,422)(268,419)(269,420)(270,418)(271,352)(272,353)(273,354)(274,360)(275,358)(276,359)(277,356)(278,357)(279,355)(280,370)(281,371)(282,372)(283,378)(284,376)(285,377)(286,374)(287,375)(288,373)(289,361)(290,362)(291,363)(292,369)(293,367)(294,368)(295,365)(296,366)(297,364)(298,325)(299,326)(300,327)(301,333)(302,331)(303,332)(304,329)(305,330)(306,328)(307,343)(308,344)(309,345)(310,351)(311,349)(312,350)(313,347)(314,348)(315,346)(316,334)(317,335)(318,336)(319,342)(320,340)(321,341)(322,338)(323,339)(324,337);
s1 := Sym(434)!( 1, 10)( 2, 12)( 3, 11)( 4, 15)( 5, 14)( 6, 13)( 7, 17)( 8, 16)( 9, 18)( 20, 21)( 22, 24)( 25, 26)( 28, 37)( 29, 39)( 30, 38)( 31, 42)( 32, 41)( 33, 40)( 34, 44)( 35, 43)( 36, 45)( 47, 48)( 49, 51)( 52, 53)( 55, 91)( 56, 93)( 57, 92)( 58, 96)( 59, 95)( 60, 94)( 61, 98)( 62, 97)( 63, 99)( 64, 82)( 65, 84)( 66, 83)( 67, 87)( 68, 86)( 69, 85)( 70, 89)( 71, 88)( 72, 90)( 73,100)( 74,102)( 75,101)( 76,105)( 77,104)( 78,103)( 79,107)( 80,106)( 81,108)(109,145)(110,147)(111,146)(112,150)(113,149)(114,148)(115,152)(116,151)(117,153)(118,136)(119,138)(120,137)(121,141)(122,140)(123,139)(124,143)(125,142)(126,144)(127,154)(128,156)(129,155)(130,159)(131,158)(132,157)(133,161)(134,160)(135,162)(163,172)(164,174)(165,173)(166,177)(167,176)(168,175)(169,179)(170,178)(171,180)(182,183)(184,186)(187,188)(190,199)(191,201)(192,200)(193,204)(194,203)(195,202)(196,206)(197,205)(198,207)(209,210)(211,213)(214,215)(217,280)(218,282)(219,281)(220,285)(221,284)(222,283)(223,287)(224,286)(225,288)(226,271)(227,273)(228,272)(229,276)(230,275)(231,274)(232,278)(233,277)(234,279)(235,289)(236,291)(237,290)(238,294)(239,293)(240,292)(241,296)(242,295)(243,297)(244,307)(245,309)(246,308)(247,312)(248,311)(249,310)(250,314)(251,313)(252,315)(253,298)(254,300)(255,299)(256,303)(257,302)(258,301)(259,305)(260,304)(261,306)(262,316)(263,318)(264,317)(265,321)(266,320)(267,319)(268,323)(269,322)(270,324)(325,415)(326,417)(327,416)(328,420)(329,419)(330,418)(331,422)(332,421)(333,423)(334,406)(335,408)(336,407)(337,411)(338,410)(339,409)(340,413)(341,412)(342,414)(343,424)(344,426)(345,425)(346,429)(347,428)(348,427)(349,431)(350,430)(351,432)(352,388)(353,390)(354,389)(355,393)(356,392)(357,391)(358,395)(359,394)(360,396)(361,379)(362,381)(363,380)(364,384)(365,383)(366,382)(367,386)(368,385)(369,387)(370,397)(371,399)(372,398)(373,402)(374,401)(375,400)(376,404)(377,403)(378,405);
s2 := Sym(434)!( 1,272)( 2,271)( 3,273)( 4,289)( 5,291)( 6,290)( 7,282)( 8,281)( 9,280)( 10,279)( 11,278)( 12,277)( 13,297)( 14,296)( 15,295)( 16,288)( 17,287)( 18,286)( 19,274)( 20,276)( 21,275)( 22,293)( 23,292)( 24,294)( 25,285)( 26,284)( 27,283)( 28,299)( 29,298)( 30,300)( 31,316)( 32,318)( 33,317)( 34,309)( 35,308)( 36,307)( 37,306)( 38,305)( 39,304)( 40,324)( 41,323)( 42,322)( 43,315)( 44,314)( 45,313)( 46,301)( 47,303)( 48,302)( 49,320)( 50,319)( 51,321)( 52,312)( 53,311)( 54,310)( 55,218)( 56,217)( 57,219)( 58,235)( 59,237)( 60,236)( 61,228)( 62,227)( 63,226)( 64,225)( 65,224)( 66,223)( 67,243)( 68,242)( 69,241)( 70,234)( 71,233)( 72,232)( 73,220)( 74,222)( 75,221)( 76,239)( 77,238)( 78,240)( 79,231)( 80,230)( 81,229)( 82,245)( 83,244)( 84,246)( 85,262)( 86,264)( 87,263)( 88,255)( 89,254)( 90,253)( 91,252)( 92,251)( 93,250)( 94,270)( 95,269)( 96,268)( 97,261)( 98,260)( 99,259)(100,247)(101,249)(102,248)(103,266)(104,265)(105,267)(106,258)(107,257)(108,256)(109,407)(110,406)(111,408)(112,424)(113,426)(114,425)(115,417)(116,416)(117,415)(118,414)(119,413)(120,412)(121,432)(122,431)(123,430)(124,423)(125,422)(126,421)(127,409)(128,411)(129,410)(130,428)(131,427)(132,429)(133,420)(134,419)(135,418)(136,380)(137,379)(138,381)(139,397)(140,399)(141,398)(142,390)(143,389)(144,388)(145,387)(146,386)(147,385)(148,405)(149,404)(150,403)(151,396)(152,395)(153,394)(154,382)(155,384)(156,383)(157,401)(158,400)(159,402)(160,393)(161,392)(162,391)(163,353)(164,352)(165,354)(166,370)(167,372)(168,371)(169,363)(170,362)(171,361)(172,360)(173,359)(174,358)(175,378)(176,377)(177,376)(178,369)(179,368)(180,367)(181,355)(182,357)(183,356)(184,374)(185,373)(186,375)(187,366)(188,365)(189,364)(190,326)(191,325)(192,327)(193,343)(194,345)(195,344)(196,336)(197,335)(198,334)(199,333)(200,332)(201,331)(202,351)(203,350)(204,349)(205,342)(206,341)(207,340)(208,328)(209,330)(210,329)(211,347)(212,346)(213,348)(214,339)(215,338)(216,337);
s3 := Sym(434)!(433,434);
poly := sub<Sym(434)|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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2 >;
to this polytope