Polytope of Type {2,54,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,54,4}*1728
if this polytope has a name.
Group : SmallGroup(1728,20782)
Rank : 4
Schlafli Type : {2,54,4}
Number of vertices, edges, etc : 2, 108, 216, 8
Order of s0s1s2s3 : 54
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 : {2,27,4}*864, {2,54,4}*864b, {2,54,4}*864c
3-fold quotients : {2,18,4}*576
4-fold quotients : {2,27,4}*432, {2,54,2}*432
6-fold quotients : {2,9,4}*288, {2,18,4}*288b, {2,18,4}*288c
8-fold quotients : {2,27,2}*216
9-fold quotients : {2,6,4}*192
12-fold quotients : {2,9,4}*144, {2,18,2}*144
18-fold quotients : {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
24-fold quotients : {2,9,2}*72
36-fold quotients : {2,3,4}*48, {2,6,2}*48
72-fold quotients : {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 15, 31)( 16, 33)( 17, 32)
( 18, 34)( 19, 27)( 20, 29)( 21, 28)( 22, 30)( 23, 35)( 24, 37)( 25, 36)
( 26, 38)( 39, 99)( 40,101)( 41,100)( 42,102)( 43,107)( 44,109)( 45,108)
( 46,110)( 47,103)( 48,105)( 49,104)( 50,106)( 51, 87)( 52, 89)( 53, 88)
( 54, 90)( 55, 95)( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)( 61, 92)
( 62, 94)( 63, 75)( 64, 77)( 65, 76)( 66, 78)( 67, 83)( 68, 85)( 69, 84)
( 70, 86)( 71, 79)( 72, 81)( 73, 80)( 74, 82)(112,113)(115,119)(116,121)
(117,120)(118,122)(123,139)(124,141)(125,140)(126,142)(127,135)(128,137)
(129,136)(130,138)(131,143)(132,145)(133,144)(134,146)(147,207)(148,209)
(149,208)(150,210)(151,215)(152,217)(153,216)(154,218)(155,211)(156,213)
(157,212)(158,214)(159,195)(160,197)(161,196)(162,198)(163,203)(164,205)
(165,204)(166,206)(167,199)(168,201)(169,200)(170,202)(171,183)(172,185)
(173,184)(174,186)(175,191)(176,193)(177,192)(178,194)(179,187)(180,189)
(181,188)(182,190)(220,221)(223,227)(224,229)(225,228)(226,230)(231,247)
(232,249)(233,248)(234,250)(235,243)(236,245)(237,244)(238,246)(239,251)
(240,253)(241,252)(242,254)(255,315)(256,317)(257,316)(258,318)(259,323)
(260,325)(261,324)(262,326)(263,319)(264,321)(265,320)(266,322)(267,303)
(268,305)(269,304)(270,306)(271,311)(272,313)(273,312)(274,314)(275,307)
(276,309)(277,308)(278,310)(279,291)(280,293)(281,292)(282,294)(283,299)
(284,301)(285,300)(286,302)(287,295)(288,297)(289,296)(290,298)(328,329)
(331,335)(332,337)(333,336)(334,338)(339,355)(340,357)(341,356)(342,358)
(343,351)(344,353)(345,352)(346,354)(347,359)(348,361)(349,360)(350,362)
(363,423)(364,425)(365,424)(366,426)(367,431)(368,433)(369,432)(370,434)
(371,427)(372,429)(373,428)(374,430)(375,411)(376,413)(377,412)(378,414)
(379,419)(380,421)(381,420)(382,422)(383,415)(384,417)(385,416)(386,418)
(387,399)(388,401)(389,400)(390,402)(391,407)(392,409)(393,408)(394,410)
(395,403)(396,405)(397,404)(398,406);;
s2 := ( 3,255)( 4,256)( 5,258)( 6,257)( 7,263)( 8,264)( 9,266)( 10,265)
( 11,259)( 12,260)( 13,262)( 14,261)( 15,283)( 16,284)( 17,286)( 18,285)
( 19,279)( 20,280)( 21,282)( 22,281)( 23,287)( 24,288)( 25,290)( 26,289)
( 27,271)( 28,272)( 29,274)( 30,273)( 31,267)( 32,268)( 33,270)( 34,269)
( 35,275)( 36,276)( 37,278)( 38,277)( 39,219)( 40,220)( 41,222)( 42,221)
( 43,227)( 44,228)( 45,230)( 46,229)( 47,223)( 48,224)( 49,226)( 50,225)
( 51,247)( 52,248)( 53,250)( 54,249)( 55,243)( 56,244)( 57,246)( 58,245)
( 59,251)( 60,252)( 61,254)( 62,253)( 63,235)( 64,236)( 65,238)( 66,237)
( 67,231)( 68,232)( 69,234)( 70,233)( 71,239)( 72,240)( 73,242)( 74,241)
( 75,315)( 76,316)( 77,318)( 78,317)( 79,323)( 80,324)( 81,326)( 82,325)
( 83,319)( 84,320)( 85,322)( 86,321)( 87,303)( 88,304)( 89,306)( 90,305)
( 91,311)( 92,312)( 93,314)( 94,313)( 95,307)( 96,308)( 97,310)( 98,309)
( 99,291)(100,292)(101,294)(102,293)(103,299)(104,300)(105,302)(106,301)
(107,295)(108,296)(109,298)(110,297)(111,363)(112,364)(113,366)(114,365)
(115,371)(116,372)(117,374)(118,373)(119,367)(120,368)(121,370)(122,369)
(123,391)(124,392)(125,394)(126,393)(127,387)(128,388)(129,390)(130,389)
(131,395)(132,396)(133,398)(134,397)(135,379)(136,380)(137,382)(138,381)
(139,375)(140,376)(141,378)(142,377)(143,383)(144,384)(145,386)(146,385)
(147,327)(148,328)(149,330)(150,329)(151,335)(152,336)(153,338)(154,337)
(155,331)(156,332)(157,334)(158,333)(159,355)(160,356)(161,358)(162,357)
(163,351)(164,352)(165,354)(166,353)(167,359)(168,360)(169,362)(170,361)
(171,343)(172,344)(173,346)(174,345)(175,339)(176,340)(177,342)(178,341)
(179,347)(180,348)(181,350)(182,349)(183,423)(184,424)(185,426)(186,425)
(187,431)(188,432)(189,434)(190,433)(191,427)(192,428)(193,430)(194,429)
(195,411)(196,412)(197,414)(198,413)(199,419)(200,420)(201,422)(202,421)
(203,415)(204,416)(205,418)(206,417)(207,399)(208,400)(209,402)(210,401)
(211,407)(212,408)(213,410)(214,409)(215,403)(216,404)(217,406)(218,405);;
s3 := ( 3,114)( 4,113)( 5,112)( 6,111)( 7,118)( 8,117)( 9,116)( 10,115)
( 11,122)( 12,121)( 13,120)( 14,119)( 15,126)( 16,125)( 17,124)( 18,123)
( 19,130)( 20,129)( 21,128)( 22,127)( 23,134)( 24,133)( 25,132)( 26,131)
( 27,138)( 28,137)( 29,136)( 30,135)( 31,142)( 32,141)( 33,140)( 34,139)
( 35,146)( 36,145)( 37,144)( 38,143)( 39,150)( 40,149)( 41,148)( 42,147)
( 43,154)( 44,153)( 45,152)( 46,151)( 47,158)( 48,157)( 49,156)( 50,155)
( 51,162)( 52,161)( 53,160)( 54,159)( 55,166)( 56,165)( 57,164)( 58,163)
( 59,170)( 60,169)( 61,168)( 62,167)( 63,174)( 64,173)( 65,172)( 66,171)
( 67,178)( 68,177)( 69,176)( 70,175)( 71,182)( 72,181)( 73,180)( 74,179)
( 75,186)( 76,185)( 77,184)( 78,183)( 79,190)( 80,189)( 81,188)( 82,187)
( 83,194)( 84,193)( 85,192)( 86,191)( 87,198)( 88,197)( 89,196)( 90,195)
( 91,202)( 92,201)( 93,200)( 94,199)( 95,206)( 96,205)( 97,204)( 98,203)
( 99,210)(100,209)(101,208)(102,207)(103,214)(104,213)(105,212)(106,211)
(107,218)(108,217)(109,216)(110,215)(219,330)(220,329)(221,328)(222,327)
(223,334)(224,333)(225,332)(226,331)(227,338)(228,337)(229,336)(230,335)
(231,342)(232,341)(233,340)(234,339)(235,346)(236,345)(237,344)(238,343)
(239,350)(240,349)(241,348)(242,347)(243,354)(244,353)(245,352)(246,351)
(247,358)(248,357)(249,356)(250,355)(251,362)(252,361)(253,360)(254,359)
(255,366)(256,365)(257,364)(258,363)(259,370)(260,369)(261,368)(262,367)
(263,374)(264,373)(265,372)(266,371)(267,378)(268,377)(269,376)(270,375)
(271,382)(272,381)(273,380)(274,379)(275,386)(276,385)(277,384)(278,383)
(279,390)(280,389)(281,388)(282,387)(283,394)(284,393)(285,392)(286,391)
(287,398)(288,397)(289,396)(290,395)(291,402)(292,401)(293,400)(294,399)
(295,406)(296,405)(297,404)(298,403)(299,410)(300,409)(301,408)(302,407)
(303,414)(304,413)(305,412)(306,411)(307,418)(308,417)(309,416)(310,415)
(311,422)(312,421)(313,420)(314,419)(315,426)(316,425)(317,424)(318,423)
(319,430)(320,429)(321,428)(322,427)(323,434)(324,433)(325,432)(326,431);;
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,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(434)!(1,2);
s1 := Sym(434)!( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 15, 31)( 16, 33)
( 17, 32)( 18, 34)( 19, 27)( 20, 29)( 21, 28)( 22, 30)( 23, 35)( 24, 37)
( 25, 36)( 26, 38)( 39, 99)( 40,101)( 41,100)( 42,102)( 43,107)( 44,109)
( 45,108)( 46,110)( 47,103)( 48,105)( 49,104)( 50,106)( 51, 87)( 52, 89)
( 53, 88)( 54, 90)( 55, 95)( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)
( 61, 92)( 62, 94)( 63, 75)( 64, 77)( 65, 76)( 66, 78)( 67, 83)( 68, 85)
( 69, 84)( 70, 86)( 71, 79)( 72, 81)( 73, 80)( 74, 82)(112,113)(115,119)
(116,121)(117,120)(118,122)(123,139)(124,141)(125,140)(126,142)(127,135)
(128,137)(129,136)(130,138)(131,143)(132,145)(133,144)(134,146)(147,207)
(148,209)(149,208)(150,210)(151,215)(152,217)(153,216)(154,218)(155,211)
(156,213)(157,212)(158,214)(159,195)(160,197)(161,196)(162,198)(163,203)
(164,205)(165,204)(166,206)(167,199)(168,201)(169,200)(170,202)(171,183)
(172,185)(173,184)(174,186)(175,191)(176,193)(177,192)(178,194)(179,187)
(180,189)(181,188)(182,190)(220,221)(223,227)(224,229)(225,228)(226,230)
(231,247)(232,249)(233,248)(234,250)(235,243)(236,245)(237,244)(238,246)
(239,251)(240,253)(241,252)(242,254)(255,315)(256,317)(257,316)(258,318)
(259,323)(260,325)(261,324)(262,326)(263,319)(264,321)(265,320)(266,322)
(267,303)(268,305)(269,304)(270,306)(271,311)(272,313)(273,312)(274,314)
(275,307)(276,309)(277,308)(278,310)(279,291)(280,293)(281,292)(282,294)
(283,299)(284,301)(285,300)(286,302)(287,295)(288,297)(289,296)(290,298)
(328,329)(331,335)(332,337)(333,336)(334,338)(339,355)(340,357)(341,356)
(342,358)(343,351)(344,353)(345,352)(346,354)(347,359)(348,361)(349,360)
(350,362)(363,423)(364,425)(365,424)(366,426)(367,431)(368,433)(369,432)
(370,434)(371,427)(372,429)(373,428)(374,430)(375,411)(376,413)(377,412)
(378,414)(379,419)(380,421)(381,420)(382,422)(383,415)(384,417)(385,416)
(386,418)(387,399)(388,401)(389,400)(390,402)(391,407)(392,409)(393,408)
(394,410)(395,403)(396,405)(397,404)(398,406);
s2 := Sym(434)!( 3,255)( 4,256)( 5,258)( 6,257)( 7,263)( 8,264)( 9,266)
( 10,265)( 11,259)( 12,260)( 13,262)( 14,261)( 15,283)( 16,284)( 17,286)
( 18,285)( 19,279)( 20,280)( 21,282)( 22,281)( 23,287)( 24,288)( 25,290)
( 26,289)( 27,271)( 28,272)( 29,274)( 30,273)( 31,267)( 32,268)( 33,270)
( 34,269)( 35,275)( 36,276)( 37,278)( 38,277)( 39,219)( 40,220)( 41,222)
( 42,221)( 43,227)( 44,228)( 45,230)( 46,229)( 47,223)( 48,224)( 49,226)
( 50,225)( 51,247)( 52,248)( 53,250)( 54,249)( 55,243)( 56,244)( 57,246)
( 58,245)( 59,251)( 60,252)( 61,254)( 62,253)( 63,235)( 64,236)( 65,238)
( 66,237)( 67,231)( 68,232)( 69,234)( 70,233)( 71,239)( 72,240)( 73,242)
( 74,241)( 75,315)( 76,316)( 77,318)( 78,317)( 79,323)( 80,324)( 81,326)
( 82,325)( 83,319)( 84,320)( 85,322)( 86,321)( 87,303)( 88,304)( 89,306)
( 90,305)( 91,311)( 92,312)( 93,314)( 94,313)( 95,307)( 96,308)( 97,310)
( 98,309)( 99,291)(100,292)(101,294)(102,293)(103,299)(104,300)(105,302)
(106,301)(107,295)(108,296)(109,298)(110,297)(111,363)(112,364)(113,366)
(114,365)(115,371)(116,372)(117,374)(118,373)(119,367)(120,368)(121,370)
(122,369)(123,391)(124,392)(125,394)(126,393)(127,387)(128,388)(129,390)
(130,389)(131,395)(132,396)(133,398)(134,397)(135,379)(136,380)(137,382)
(138,381)(139,375)(140,376)(141,378)(142,377)(143,383)(144,384)(145,386)
(146,385)(147,327)(148,328)(149,330)(150,329)(151,335)(152,336)(153,338)
(154,337)(155,331)(156,332)(157,334)(158,333)(159,355)(160,356)(161,358)
(162,357)(163,351)(164,352)(165,354)(166,353)(167,359)(168,360)(169,362)
(170,361)(171,343)(172,344)(173,346)(174,345)(175,339)(176,340)(177,342)
(178,341)(179,347)(180,348)(181,350)(182,349)(183,423)(184,424)(185,426)
(186,425)(187,431)(188,432)(189,434)(190,433)(191,427)(192,428)(193,430)
(194,429)(195,411)(196,412)(197,414)(198,413)(199,419)(200,420)(201,422)
(202,421)(203,415)(204,416)(205,418)(206,417)(207,399)(208,400)(209,402)
(210,401)(211,407)(212,408)(213,410)(214,409)(215,403)(216,404)(217,406)
(218,405);
s3 := Sym(434)!( 3,114)( 4,113)( 5,112)( 6,111)( 7,118)( 8,117)( 9,116)
( 10,115)( 11,122)( 12,121)( 13,120)( 14,119)( 15,126)( 16,125)( 17,124)
( 18,123)( 19,130)( 20,129)( 21,128)( 22,127)( 23,134)( 24,133)( 25,132)
( 26,131)( 27,138)( 28,137)( 29,136)( 30,135)( 31,142)( 32,141)( 33,140)
( 34,139)( 35,146)( 36,145)( 37,144)( 38,143)( 39,150)( 40,149)( 41,148)
( 42,147)( 43,154)( 44,153)( 45,152)( 46,151)( 47,158)( 48,157)( 49,156)
( 50,155)( 51,162)( 52,161)( 53,160)( 54,159)( 55,166)( 56,165)( 57,164)
( 58,163)( 59,170)( 60,169)( 61,168)( 62,167)( 63,174)( 64,173)( 65,172)
( 66,171)( 67,178)( 68,177)( 69,176)( 70,175)( 71,182)( 72,181)( 73,180)
( 74,179)( 75,186)( 76,185)( 77,184)( 78,183)( 79,190)( 80,189)( 81,188)
( 82,187)( 83,194)( 84,193)( 85,192)( 86,191)( 87,198)( 88,197)( 89,196)
( 90,195)( 91,202)( 92,201)( 93,200)( 94,199)( 95,206)( 96,205)( 97,204)
( 98,203)( 99,210)(100,209)(101,208)(102,207)(103,214)(104,213)(105,212)
(106,211)(107,218)(108,217)(109,216)(110,215)(219,330)(220,329)(221,328)
(222,327)(223,334)(224,333)(225,332)(226,331)(227,338)(228,337)(229,336)
(230,335)(231,342)(232,341)(233,340)(234,339)(235,346)(236,345)(237,344)
(238,343)(239,350)(240,349)(241,348)(242,347)(243,354)(244,353)(245,352)
(246,351)(247,358)(248,357)(249,356)(250,355)(251,362)(252,361)(253,360)
(254,359)(255,366)(256,365)(257,364)(258,363)(259,370)(260,369)(261,368)
(262,367)(263,374)(264,373)(265,372)(266,371)(267,378)(268,377)(269,376)
(270,375)(271,382)(272,381)(273,380)(274,379)(275,386)(276,385)(277,384)
(278,383)(279,390)(280,389)(281,388)(282,387)(283,394)(284,393)(285,392)
(286,391)(287,398)(288,397)(289,396)(290,395)(291,402)(292,401)(293,400)
(294,399)(295,406)(296,405)(297,404)(298,403)(299,410)(300,409)(301,408)
(302,407)(303,414)(304,413)(305,412)(306,411)(307,418)(308,417)(309,416)
(310,415)(311,422)(312,421)(313,420)(314,419)(315,426)(316,425)(317,424)
(318,423)(319,430)(320,429)(321,428)(322,427)(323,434)(324,433)(325,432)
(326,431);
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, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope