Polytope of Type {2,6,24}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,24}*1728h
if this polytope has a name.
Group : SmallGroup(1728,47196)
Rank : 4
Schlafli Type : {2,6,24}
Number of vertices, edges, etc : 2, 18, 216, 72
Order of s0s1s2s3 : 24
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 : {2,6,12}*864i
3-fold quotients : {2,6,8}*576
4-fold quotients : {2,6,12}*432c
6-fold quotients : {2,6,4}*288
9-fold quotients : {2,2,24}*192
12-fold quotients : {2,6,4}*144
18-fold quotients : {2,2,12}*96
27-fold quotients : {2,2,8}*64
36-fold quotients : {2,2,6}*48
54-fold quotients : {2,2,4}*32
72-fold quotients : {2,2,3}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,165)( 4,166)( 5,167)( 6,171)( 7,172)( 8,173)( 9,168)( 10,169)
( 11,170)( 12,183)( 13,184)( 14,185)( 15,189)( 16,190)( 17,191)( 18,186)
( 19,187)( 20,188)( 21,174)( 22,175)( 23,176)( 24,180)( 25,181)( 26,182)
( 27,177)( 28,178)( 29,179)( 30,192)( 31,193)( 32,194)( 33,198)( 34,199)
( 35,200)( 36,195)( 37,196)( 38,197)( 39,210)( 40,211)( 41,212)( 42,216)
( 43,217)( 44,218)( 45,213)( 46,214)( 47,215)( 48,201)( 49,202)( 50,203)
( 51,207)( 52,208)( 53,209)( 54,204)( 55,205)( 56,206)( 57,138)( 58,139)
( 59,140)( 60,144)( 61,145)( 62,146)( 63,141)( 64,142)( 65,143)( 66,156)
( 67,157)( 68,158)( 69,162)( 70,163)( 71,164)( 72,159)( 73,160)( 74,161)
( 75,147)( 76,148)( 77,149)( 78,153)( 79,154)( 80,155)( 81,150)( 82,151)
( 83,152)( 84,111)( 85,112)( 86,113)( 87,117)( 88,118)( 89,119)( 90,114)
( 91,115)( 92,116)( 93,129)( 94,130)( 95,131)( 96,135)( 97,136)( 98,137)
( 99,132)(100,133)(101,134)(102,120)(103,121)(104,122)(105,126)(106,127)
(107,128)(108,123)(109,124)(110,125)(219,381)(220,382)(221,383)(222,387)
(223,388)(224,389)(225,384)(226,385)(227,386)(228,399)(229,400)(230,401)
(231,405)(232,406)(233,407)(234,402)(235,403)(236,404)(237,390)(238,391)
(239,392)(240,396)(241,397)(242,398)(243,393)(244,394)(245,395)(246,408)
(247,409)(248,410)(249,414)(250,415)(251,416)(252,411)(253,412)(254,413)
(255,426)(256,427)(257,428)(258,432)(259,433)(260,434)(261,429)(262,430)
(263,431)(264,417)(265,418)(266,419)(267,423)(268,424)(269,425)(270,420)
(271,421)(272,422)(273,354)(274,355)(275,356)(276,360)(277,361)(278,362)
(279,357)(280,358)(281,359)(282,372)(283,373)(284,374)(285,378)(286,379)
(287,380)(288,375)(289,376)(290,377)(291,363)(292,364)(293,365)(294,369)
(295,370)(296,371)(297,366)(298,367)(299,368)(300,327)(301,328)(302,329)
(303,333)(304,334)(305,335)(306,330)(307,331)(308,332)(309,345)(310,346)
(311,347)(312,351)(313,352)(314,353)(315,348)(316,349)(317,350)(318,336)
(319,337)(320,338)(321,342)(322,343)(323,344)(324,339)(325,340)(326,341);;
s2 := ( 3, 12)( 4, 14)( 5, 13)( 7, 8)( 9, 27)( 10, 29)( 11, 28)( 15, 24)
( 16, 26)( 17, 25)( 19, 20)( 22, 23)( 30, 39)( 31, 41)( 32, 40)( 34, 35)
( 36, 54)( 37, 56)( 38, 55)( 42, 51)( 43, 53)( 44, 52)( 46, 47)( 49, 50)
( 57, 93)( 58, 95)( 59, 94)( 60, 87)( 61, 89)( 62, 88)( 63,108)( 64,110)
( 65,109)( 66, 84)( 67, 86)( 68, 85)( 69,105)( 70,107)( 71,106)( 72, 99)
( 73,101)( 74,100)( 75,102)( 76,104)( 77,103)( 78, 96)( 79, 98)( 80, 97)
( 81, 90)( 82, 92)( 83, 91)(111,147)(112,149)(113,148)(114,141)(115,143)
(116,142)(117,162)(118,164)(119,163)(120,138)(121,140)(122,139)(123,159)
(124,161)(125,160)(126,153)(127,155)(128,154)(129,156)(130,158)(131,157)
(132,150)(133,152)(134,151)(135,144)(136,146)(137,145)(165,174)(166,176)
(167,175)(169,170)(171,189)(172,191)(173,190)(177,186)(178,188)(179,187)
(181,182)(184,185)(192,201)(193,203)(194,202)(196,197)(198,216)(199,218)
(200,217)(204,213)(205,215)(206,214)(208,209)(211,212)(219,282)(220,284)
(221,283)(222,276)(223,278)(224,277)(225,297)(226,299)(227,298)(228,273)
(229,275)(230,274)(231,294)(232,296)(233,295)(234,288)(235,290)(236,289)
(237,291)(238,293)(239,292)(240,285)(241,287)(242,286)(243,279)(244,281)
(245,280)(246,309)(247,311)(248,310)(249,303)(250,305)(251,304)(252,324)
(253,326)(254,325)(255,300)(256,302)(257,301)(258,321)(259,323)(260,322)
(261,315)(262,317)(263,316)(264,318)(265,320)(266,319)(267,312)(268,314)
(269,313)(270,306)(271,308)(272,307)(327,417)(328,419)(329,418)(330,411)
(331,413)(332,412)(333,432)(334,434)(335,433)(336,408)(337,410)(338,409)
(339,429)(340,431)(341,430)(342,423)(343,425)(344,424)(345,426)(346,428)
(347,427)(348,420)(349,422)(350,421)(351,414)(352,416)(353,415)(354,390)
(355,392)(356,391)(357,384)(358,386)(359,385)(360,405)(361,407)(362,406)
(363,381)(364,383)(365,382)(366,402)(367,404)(368,403)(369,396)(370,398)
(371,397)(372,399)(373,401)(374,400)(375,393)(376,395)(377,394)(378,387)
(379,389)(380,388);;
s3 := ( 3,220)( 4,219)( 5,221)( 6,226)( 7,225)( 8,227)( 9,223)( 10,222)
( 11,224)( 12,232)( 13,231)( 14,233)( 15,229)( 16,228)( 17,230)( 18,235)
( 19,234)( 20,236)( 21,244)( 22,243)( 23,245)( 24,241)( 25,240)( 26,242)
( 27,238)( 28,237)( 29,239)( 30,247)( 31,246)( 32,248)( 33,253)( 34,252)
( 35,254)( 36,250)( 37,249)( 38,251)( 39,259)( 40,258)( 41,260)( 42,256)
( 43,255)( 44,257)( 45,262)( 46,261)( 47,263)( 48,271)( 49,270)( 50,272)
( 51,268)( 52,267)( 53,269)( 54,265)( 55,264)( 56,266)( 57,301)( 58,300)
( 59,302)( 60,307)( 61,306)( 62,308)( 63,304)( 64,303)( 65,305)( 66,313)
( 67,312)( 68,314)( 69,310)( 70,309)( 71,311)( 72,316)( 73,315)( 74,317)
( 75,325)( 76,324)( 77,326)( 78,322)( 79,321)( 80,323)( 81,319)( 82,318)
( 83,320)( 84,274)( 85,273)( 86,275)( 87,280)( 88,279)( 89,281)( 90,277)
( 91,276)( 92,278)( 93,286)( 94,285)( 95,287)( 96,283)( 97,282)( 98,284)
( 99,289)(100,288)(101,290)(102,298)(103,297)(104,299)(105,295)(106,294)
(107,296)(108,292)(109,291)(110,293)(111,355)(112,354)(113,356)(114,361)
(115,360)(116,362)(117,358)(118,357)(119,359)(120,367)(121,366)(122,368)
(123,364)(124,363)(125,365)(126,370)(127,369)(128,371)(129,379)(130,378)
(131,380)(132,376)(133,375)(134,377)(135,373)(136,372)(137,374)(138,328)
(139,327)(140,329)(141,334)(142,333)(143,335)(144,331)(145,330)(146,332)
(147,340)(148,339)(149,341)(150,337)(151,336)(152,338)(153,343)(154,342)
(155,344)(156,352)(157,351)(158,353)(159,349)(160,348)(161,350)(162,346)
(163,345)(164,347)(165,382)(166,381)(167,383)(168,388)(169,387)(170,389)
(171,385)(172,384)(173,386)(174,394)(175,393)(176,395)(177,391)(178,390)
(179,392)(180,397)(181,396)(182,398)(183,406)(184,405)(185,407)(186,403)
(187,402)(188,404)(189,400)(190,399)(191,401)(192,409)(193,408)(194,410)
(195,415)(196,414)(197,416)(198,412)(199,411)(200,413)(201,421)(202,420)
(203,422)(204,418)(205,417)(206,419)(207,424)(208,423)(209,425)(210,433)
(211,432)(212,434)(213,430)(214,429)(215,431)(216,427)(217,426)(218,428);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2*s1*s2,
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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(434)!(1,2);
s1 := Sym(434)!( 3,165)( 4,166)( 5,167)( 6,171)( 7,172)( 8,173)( 9,168)
( 10,169)( 11,170)( 12,183)( 13,184)( 14,185)( 15,189)( 16,190)( 17,191)
( 18,186)( 19,187)( 20,188)( 21,174)( 22,175)( 23,176)( 24,180)( 25,181)
( 26,182)( 27,177)( 28,178)( 29,179)( 30,192)( 31,193)( 32,194)( 33,198)
( 34,199)( 35,200)( 36,195)( 37,196)( 38,197)( 39,210)( 40,211)( 41,212)
( 42,216)( 43,217)( 44,218)( 45,213)( 46,214)( 47,215)( 48,201)( 49,202)
( 50,203)( 51,207)( 52,208)( 53,209)( 54,204)( 55,205)( 56,206)( 57,138)
( 58,139)( 59,140)( 60,144)( 61,145)( 62,146)( 63,141)( 64,142)( 65,143)
( 66,156)( 67,157)( 68,158)( 69,162)( 70,163)( 71,164)( 72,159)( 73,160)
( 74,161)( 75,147)( 76,148)( 77,149)( 78,153)( 79,154)( 80,155)( 81,150)
( 82,151)( 83,152)( 84,111)( 85,112)( 86,113)( 87,117)( 88,118)( 89,119)
( 90,114)( 91,115)( 92,116)( 93,129)( 94,130)( 95,131)( 96,135)( 97,136)
( 98,137)( 99,132)(100,133)(101,134)(102,120)(103,121)(104,122)(105,126)
(106,127)(107,128)(108,123)(109,124)(110,125)(219,381)(220,382)(221,383)
(222,387)(223,388)(224,389)(225,384)(226,385)(227,386)(228,399)(229,400)
(230,401)(231,405)(232,406)(233,407)(234,402)(235,403)(236,404)(237,390)
(238,391)(239,392)(240,396)(241,397)(242,398)(243,393)(244,394)(245,395)
(246,408)(247,409)(248,410)(249,414)(250,415)(251,416)(252,411)(253,412)
(254,413)(255,426)(256,427)(257,428)(258,432)(259,433)(260,434)(261,429)
(262,430)(263,431)(264,417)(265,418)(266,419)(267,423)(268,424)(269,425)
(270,420)(271,421)(272,422)(273,354)(274,355)(275,356)(276,360)(277,361)
(278,362)(279,357)(280,358)(281,359)(282,372)(283,373)(284,374)(285,378)
(286,379)(287,380)(288,375)(289,376)(290,377)(291,363)(292,364)(293,365)
(294,369)(295,370)(296,371)(297,366)(298,367)(299,368)(300,327)(301,328)
(302,329)(303,333)(304,334)(305,335)(306,330)(307,331)(308,332)(309,345)
(310,346)(311,347)(312,351)(313,352)(314,353)(315,348)(316,349)(317,350)
(318,336)(319,337)(320,338)(321,342)(322,343)(323,344)(324,339)(325,340)
(326,341);
s2 := Sym(434)!( 3, 12)( 4, 14)( 5, 13)( 7, 8)( 9, 27)( 10, 29)( 11, 28)
( 15, 24)( 16, 26)( 17, 25)( 19, 20)( 22, 23)( 30, 39)( 31, 41)( 32, 40)
( 34, 35)( 36, 54)( 37, 56)( 38, 55)( 42, 51)( 43, 53)( 44, 52)( 46, 47)
( 49, 50)( 57, 93)( 58, 95)( 59, 94)( 60, 87)( 61, 89)( 62, 88)( 63,108)
( 64,110)( 65,109)( 66, 84)( 67, 86)( 68, 85)( 69,105)( 70,107)( 71,106)
( 72, 99)( 73,101)( 74,100)( 75,102)( 76,104)( 77,103)( 78, 96)( 79, 98)
( 80, 97)( 81, 90)( 82, 92)( 83, 91)(111,147)(112,149)(113,148)(114,141)
(115,143)(116,142)(117,162)(118,164)(119,163)(120,138)(121,140)(122,139)
(123,159)(124,161)(125,160)(126,153)(127,155)(128,154)(129,156)(130,158)
(131,157)(132,150)(133,152)(134,151)(135,144)(136,146)(137,145)(165,174)
(166,176)(167,175)(169,170)(171,189)(172,191)(173,190)(177,186)(178,188)
(179,187)(181,182)(184,185)(192,201)(193,203)(194,202)(196,197)(198,216)
(199,218)(200,217)(204,213)(205,215)(206,214)(208,209)(211,212)(219,282)
(220,284)(221,283)(222,276)(223,278)(224,277)(225,297)(226,299)(227,298)
(228,273)(229,275)(230,274)(231,294)(232,296)(233,295)(234,288)(235,290)
(236,289)(237,291)(238,293)(239,292)(240,285)(241,287)(242,286)(243,279)
(244,281)(245,280)(246,309)(247,311)(248,310)(249,303)(250,305)(251,304)
(252,324)(253,326)(254,325)(255,300)(256,302)(257,301)(258,321)(259,323)
(260,322)(261,315)(262,317)(263,316)(264,318)(265,320)(266,319)(267,312)
(268,314)(269,313)(270,306)(271,308)(272,307)(327,417)(328,419)(329,418)
(330,411)(331,413)(332,412)(333,432)(334,434)(335,433)(336,408)(337,410)
(338,409)(339,429)(340,431)(341,430)(342,423)(343,425)(344,424)(345,426)
(346,428)(347,427)(348,420)(349,422)(350,421)(351,414)(352,416)(353,415)
(354,390)(355,392)(356,391)(357,384)(358,386)(359,385)(360,405)(361,407)
(362,406)(363,381)(364,383)(365,382)(366,402)(367,404)(368,403)(369,396)
(370,398)(371,397)(372,399)(373,401)(374,400)(375,393)(376,395)(377,394)
(378,387)(379,389)(380,388);
s3 := Sym(434)!( 3,220)( 4,219)( 5,221)( 6,226)( 7,225)( 8,227)( 9,223)
( 10,222)( 11,224)( 12,232)( 13,231)( 14,233)( 15,229)( 16,228)( 17,230)
( 18,235)( 19,234)( 20,236)( 21,244)( 22,243)( 23,245)( 24,241)( 25,240)
( 26,242)( 27,238)( 28,237)( 29,239)( 30,247)( 31,246)( 32,248)( 33,253)
( 34,252)( 35,254)( 36,250)( 37,249)( 38,251)( 39,259)( 40,258)( 41,260)
( 42,256)( 43,255)( 44,257)( 45,262)( 46,261)( 47,263)( 48,271)( 49,270)
( 50,272)( 51,268)( 52,267)( 53,269)( 54,265)( 55,264)( 56,266)( 57,301)
( 58,300)( 59,302)( 60,307)( 61,306)( 62,308)( 63,304)( 64,303)( 65,305)
( 66,313)( 67,312)( 68,314)( 69,310)( 70,309)( 71,311)( 72,316)( 73,315)
( 74,317)( 75,325)( 76,324)( 77,326)( 78,322)( 79,321)( 80,323)( 81,319)
( 82,318)( 83,320)( 84,274)( 85,273)( 86,275)( 87,280)( 88,279)( 89,281)
( 90,277)( 91,276)( 92,278)( 93,286)( 94,285)( 95,287)( 96,283)( 97,282)
( 98,284)( 99,289)(100,288)(101,290)(102,298)(103,297)(104,299)(105,295)
(106,294)(107,296)(108,292)(109,291)(110,293)(111,355)(112,354)(113,356)
(114,361)(115,360)(116,362)(117,358)(118,357)(119,359)(120,367)(121,366)
(122,368)(123,364)(124,363)(125,365)(126,370)(127,369)(128,371)(129,379)
(130,378)(131,380)(132,376)(133,375)(134,377)(135,373)(136,372)(137,374)
(138,328)(139,327)(140,329)(141,334)(142,333)(143,335)(144,331)(145,330)
(146,332)(147,340)(148,339)(149,341)(150,337)(151,336)(152,338)(153,343)
(154,342)(155,344)(156,352)(157,351)(158,353)(159,349)(160,348)(161,350)
(162,346)(163,345)(164,347)(165,382)(166,381)(167,383)(168,388)(169,387)
(170,389)(171,385)(172,384)(173,386)(174,394)(175,393)(176,395)(177,391)
(178,390)(179,392)(180,397)(181,396)(182,398)(183,406)(184,405)(185,407)
(186,403)(187,402)(188,404)(189,400)(190,399)(191,401)(192,409)(193,408)
(194,410)(195,415)(196,414)(197,416)(198,412)(199,411)(200,413)(201,421)
(202,420)(203,422)(204,418)(205,417)(206,419)(207,424)(208,423)(209,425)
(210,433)(211,432)(212,434)(213,430)(214,429)(215,431)(216,427)(217,426)
(218,428);
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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2*s1*s2,
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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope