Polytope of Type {2,4,54}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,54}*1728
if this polytope has a name.
Group : SmallGroup(1728,20782)
Rank : 4
Schlafli Type : {2,4,54}
Number of vertices, edges, etc : 2, 8, 216, 108
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 : {2,4,27}*864, {2,4,54}*864b, {2,4,54}*864c
3-fold quotients : {2,4,18}*576
4-fold quotients : {2,4,27}*432, {2,2,54}*432
6-fold quotients : {2,4,9}*288, {2,4,18}*288b, {2,4,18}*288c
8-fold quotients : {2,2,27}*216
9-fold quotients : {2,4,6}*192
12-fold quotients : {2,4,9}*144, {2,2,18}*144
18-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
24-fold quotients : {2,2,9}*72
36-fold quotients : {2,4,3}*48, {2,2,6}*48
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 := ( 3,221)( 4,222)( 5,219)( 6,220)( 7,225)( 8,226)( 9,223)( 10,224)
( 11,229)( 12,230)( 13,227)( 14,228)( 15,233)( 16,234)( 17,231)( 18,232)
( 19,237)( 20,238)( 21,235)( 22,236)( 23,241)( 24,242)( 25,239)( 26,240)
( 27,245)( 28,246)( 29,243)( 30,244)( 31,249)( 32,250)( 33,247)( 34,248)
( 35,253)( 36,254)( 37,251)( 38,252)( 39,257)( 40,258)( 41,255)( 42,256)
( 43,261)( 44,262)( 45,259)( 46,260)( 47,265)( 48,266)( 49,263)( 50,264)
( 51,269)( 52,270)( 53,267)( 54,268)( 55,273)( 56,274)( 57,271)( 58,272)
( 59,277)( 60,278)( 61,275)( 62,276)( 63,281)( 64,282)( 65,279)( 66,280)
( 67,285)( 68,286)( 69,283)( 70,284)( 71,289)( 72,290)( 73,287)( 74,288)
( 75,293)( 76,294)( 77,291)( 78,292)( 79,297)( 80,298)( 81,295)( 82,296)
( 83,301)( 84,302)( 85,299)( 86,300)( 87,305)( 88,306)( 89,303)( 90,304)
( 91,309)( 92,310)( 93,307)( 94,308)( 95,313)( 96,314)( 97,311)( 98,312)
( 99,317)(100,318)(101,315)(102,316)(103,321)(104,322)(105,319)(106,320)
(107,325)(108,326)(109,323)(110,324)(111,329)(112,330)(113,327)(114,328)
(115,333)(116,334)(117,331)(118,332)(119,337)(120,338)(121,335)(122,336)
(123,341)(124,342)(125,339)(126,340)(127,345)(128,346)(129,343)(130,344)
(131,349)(132,350)(133,347)(134,348)(135,353)(136,354)(137,351)(138,352)
(139,357)(140,358)(141,355)(142,356)(143,361)(144,362)(145,359)(146,360)
(147,365)(148,366)(149,363)(150,364)(151,369)(152,370)(153,367)(154,368)
(155,373)(156,374)(157,371)(158,372)(159,377)(160,378)(161,375)(162,376)
(163,381)(164,382)(165,379)(166,380)(167,385)(168,386)(169,383)(170,384)
(171,389)(172,390)(173,387)(174,388)(175,393)(176,394)(177,391)(178,392)
(179,397)(180,398)(181,395)(182,396)(183,401)(184,402)(185,399)(186,400)
(187,405)(188,406)(189,403)(190,404)(191,409)(192,410)(193,407)(194,408)
(195,413)(196,414)(197,411)(198,412)(199,417)(200,418)(201,415)(202,416)
(203,421)(204,422)(205,419)(206,420)(207,425)(208,426)(209,423)(210,424)
(211,429)(212,430)(213,427)(214,428)(215,433)(216,434)(217,431)(218,432);;
s2 := ( 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);;
s3 := ( 3,183)( 4,186)( 5,185)( 6,184)( 7,191)( 8,194)( 9,193)( 10,192)
( 11,187)( 12,190)( 13,189)( 14,188)( 15,211)( 16,214)( 17,213)( 18,212)
( 19,207)( 20,210)( 21,209)( 22,208)( 23,215)( 24,218)( 25,217)( 26,216)
( 27,199)( 28,202)( 29,201)( 30,200)( 31,195)( 32,198)( 33,197)( 34,196)
( 35,203)( 36,206)( 37,205)( 38,204)( 39,147)( 40,150)( 41,149)( 42,148)
( 43,155)( 44,158)( 45,157)( 46,156)( 47,151)( 48,154)( 49,153)( 50,152)
( 51,175)( 52,178)( 53,177)( 54,176)( 55,171)( 56,174)( 57,173)( 58,172)
( 59,179)( 60,182)( 61,181)( 62,180)( 63,163)( 64,166)( 65,165)( 66,164)
( 67,159)( 68,162)( 69,161)( 70,160)( 71,167)( 72,170)( 73,169)( 74,168)
( 75,111)( 76,114)( 77,113)( 78,112)( 79,119)( 80,122)( 81,121)( 82,120)
( 83,115)( 84,118)( 85,117)( 86,116)( 87,139)( 88,142)( 89,141)( 90,140)
( 91,135)( 92,138)( 93,137)( 94,136)( 95,143)( 96,146)( 97,145)( 98,144)
( 99,127)(100,130)(101,129)(102,128)(103,123)(104,126)(105,125)(106,124)
(107,131)(108,134)(109,133)(110,132)(219,399)(220,402)(221,401)(222,400)
(223,407)(224,410)(225,409)(226,408)(227,403)(228,406)(229,405)(230,404)
(231,427)(232,430)(233,429)(234,428)(235,423)(236,426)(237,425)(238,424)
(239,431)(240,434)(241,433)(242,432)(243,415)(244,418)(245,417)(246,416)
(247,411)(248,414)(249,413)(250,412)(251,419)(252,422)(253,421)(254,420)
(255,363)(256,366)(257,365)(258,364)(259,371)(260,374)(261,373)(262,372)
(263,367)(264,370)(265,369)(266,368)(267,391)(268,394)(269,393)(270,392)
(271,387)(272,390)(273,389)(274,388)(275,395)(276,398)(277,397)(278,396)
(279,379)(280,382)(281,381)(282,380)(283,375)(284,378)(285,377)(286,376)
(287,383)(288,386)(289,385)(290,384)(291,327)(292,330)(293,329)(294,328)
(295,335)(296,338)(297,337)(298,336)(299,331)(300,334)(301,333)(302,332)
(303,355)(304,358)(305,357)(306,356)(307,351)(308,354)(309,353)(310,352)
(311,359)(312,362)(313,361)(314,360)(315,343)(316,346)(317,345)(318,344)
(319,339)(320,342)(321,341)(322,340)(323,347)(324,350)(325,349)(326,348);;
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*s3*s2*s3*s2*s1*s2*s3*s2*s3*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*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*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)!( 3,221)( 4,222)( 5,219)( 6,220)( 7,225)( 8,226)( 9,223)
( 10,224)( 11,229)( 12,230)( 13,227)( 14,228)( 15,233)( 16,234)( 17,231)
( 18,232)( 19,237)( 20,238)( 21,235)( 22,236)( 23,241)( 24,242)( 25,239)
( 26,240)( 27,245)( 28,246)( 29,243)( 30,244)( 31,249)( 32,250)( 33,247)
( 34,248)( 35,253)( 36,254)( 37,251)( 38,252)( 39,257)( 40,258)( 41,255)
( 42,256)( 43,261)( 44,262)( 45,259)( 46,260)( 47,265)( 48,266)( 49,263)
( 50,264)( 51,269)( 52,270)( 53,267)( 54,268)( 55,273)( 56,274)( 57,271)
( 58,272)( 59,277)( 60,278)( 61,275)( 62,276)( 63,281)( 64,282)( 65,279)
( 66,280)( 67,285)( 68,286)( 69,283)( 70,284)( 71,289)( 72,290)( 73,287)
( 74,288)( 75,293)( 76,294)( 77,291)( 78,292)( 79,297)( 80,298)( 81,295)
( 82,296)( 83,301)( 84,302)( 85,299)( 86,300)( 87,305)( 88,306)( 89,303)
( 90,304)( 91,309)( 92,310)( 93,307)( 94,308)( 95,313)( 96,314)( 97,311)
( 98,312)( 99,317)(100,318)(101,315)(102,316)(103,321)(104,322)(105,319)
(106,320)(107,325)(108,326)(109,323)(110,324)(111,329)(112,330)(113,327)
(114,328)(115,333)(116,334)(117,331)(118,332)(119,337)(120,338)(121,335)
(122,336)(123,341)(124,342)(125,339)(126,340)(127,345)(128,346)(129,343)
(130,344)(131,349)(132,350)(133,347)(134,348)(135,353)(136,354)(137,351)
(138,352)(139,357)(140,358)(141,355)(142,356)(143,361)(144,362)(145,359)
(146,360)(147,365)(148,366)(149,363)(150,364)(151,369)(152,370)(153,367)
(154,368)(155,373)(156,374)(157,371)(158,372)(159,377)(160,378)(161,375)
(162,376)(163,381)(164,382)(165,379)(166,380)(167,385)(168,386)(169,383)
(170,384)(171,389)(172,390)(173,387)(174,388)(175,393)(176,394)(177,391)
(178,392)(179,397)(180,398)(181,395)(182,396)(183,401)(184,402)(185,399)
(186,400)(187,405)(188,406)(189,403)(190,404)(191,409)(192,410)(193,407)
(194,408)(195,413)(196,414)(197,411)(198,412)(199,417)(200,418)(201,415)
(202,416)(203,421)(204,422)(205,419)(206,420)(207,425)(208,426)(209,423)
(210,424)(211,429)(212,430)(213,427)(214,428)(215,433)(216,434)(217,431)
(218,432);
s2 := 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);
s3 := Sym(434)!( 3,183)( 4,186)( 5,185)( 6,184)( 7,191)( 8,194)( 9,193)
( 10,192)( 11,187)( 12,190)( 13,189)( 14,188)( 15,211)( 16,214)( 17,213)
( 18,212)( 19,207)( 20,210)( 21,209)( 22,208)( 23,215)( 24,218)( 25,217)
( 26,216)( 27,199)( 28,202)( 29,201)( 30,200)( 31,195)( 32,198)( 33,197)
( 34,196)( 35,203)( 36,206)( 37,205)( 38,204)( 39,147)( 40,150)( 41,149)
( 42,148)( 43,155)( 44,158)( 45,157)( 46,156)( 47,151)( 48,154)( 49,153)
( 50,152)( 51,175)( 52,178)( 53,177)( 54,176)( 55,171)( 56,174)( 57,173)
( 58,172)( 59,179)( 60,182)( 61,181)( 62,180)( 63,163)( 64,166)( 65,165)
( 66,164)( 67,159)( 68,162)( 69,161)( 70,160)( 71,167)( 72,170)( 73,169)
( 74,168)( 75,111)( 76,114)( 77,113)( 78,112)( 79,119)( 80,122)( 81,121)
( 82,120)( 83,115)( 84,118)( 85,117)( 86,116)( 87,139)( 88,142)( 89,141)
( 90,140)( 91,135)( 92,138)( 93,137)( 94,136)( 95,143)( 96,146)( 97,145)
( 98,144)( 99,127)(100,130)(101,129)(102,128)(103,123)(104,126)(105,125)
(106,124)(107,131)(108,134)(109,133)(110,132)(219,399)(220,402)(221,401)
(222,400)(223,407)(224,410)(225,409)(226,408)(227,403)(228,406)(229,405)
(230,404)(231,427)(232,430)(233,429)(234,428)(235,423)(236,426)(237,425)
(238,424)(239,431)(240,434)(241,433)(242,432)(243,415)(244,418)(245,417)
(246,416)(247,411)(248,414)(249,413)(250,412)(251,419)(252,422)(253,421)
(254,420)(255,363)(256,366)(257,365)(258,364)(259,371)(260,374)(261,373)
(262,372)(263,367)(264,370)(265,369)(266,368)(267,391)(268,394)(269,393)
(270,392)(271,387)(272,390)(273,389)(274,388)(275,395)(276,398)(277,397)
(278,396)(279,379)(280,382)(281,381)(282,380)(283,375)(284,378)(285,377)
(286,376)(287,383)(288,386)(289,385)(290,384)(291,327)(292,330)(293,329)
(294,328)(295,335)(296,338)(297,337)(298,336)(299,331)(300,334)(301,333)
(302,332)(303,355)(304,358)(305,357)(306,356)(307,351)(308,354)(309,353)
(310,352)(311,359)(312,362)(313,361)(314,360)(315,343)(316,346)(317,345)
(318,344)(319,339)(320,342)(321,341)(322,340)(323,347)(324,350)(325,349)
(326,348);
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*s3*s2*s3*s2*s1*s2*s3*s2*s3*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*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope