Polytope of Type {6,144}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,144}*1728b
if this polytope has a name.
Group : SmallGroup(1728,3030)
Rank : 3
Schlafli Type : {6,144}
Number of vertices, edges, etc : 6, 432, 144
Order of s0s1s2 : 144
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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 : {6,72}*864b
3-fold quotients : {2,144}*576, {6,48}*576b
4-fold quotients : {6,36}*432b
6-fold quotients : {2,72}*288, {6,24}*288b
8-fold quotients : {6,18}*216b
9-fold quotients : {2,48}*192
12-fold quotients : {2,36}*144, {6,12}*144b
16-fold quotients : {6,9}*108
18-fold quotients : {2,24}*96
24-fold quotients : {2,18}*72, {6,6}*72b
27-fold quotients : {2,16}*64
36-fold quotients : {2,12}*48
48-fold quotients : {2,9}*36, {6,3}*36
54-fold quotients : {2,8}*32
72-fold quotients : {2,6}*24
108-fold quotients : {2,4}*16
144-fold quotients : {2,3}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)(121,130)
(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)(147,156)
(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(172,181)(173,182)
(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189)(199,208)
(200,209)(201,210)(202,211)(203,212)(204,213)(205,214)(206,215)(207,216)
(226,235)(227,236)(228,237)(229,238)(230,239)(231,240)(232,241)(233,242)
(234,243)(253,262)(254,263)(255,264)(256,265)(257,266)(258,267)(259,268)
(260,269)(261,270)(280,289)(281,290)(282,291)(283,292)(284,293)(285,294)
(286,295)(287,296)(288,297)(307,316)(308,317)(309,318)(310,319)(311,320)
(312,321)(313,322)(314,323)(315,324)(334,343)(335,344)(336,345)(337,346)
(338,347)(339,348)(340,349)(341,350)(342,351)(361,370)(362,371)(363,372)
(364,373)(365,374)(366,375)(367,376)(368,377)(369,378)(388,397)(389,398)
(390,399)(391,400)(392,401)(393,402)(394,403)(395,404)(396,405)(415,424)
(416,425)(417,426)(418,427)(419,428)(420,429)(421,430)(422,431)(423,432);;
s1 := ( 1, 10)( 2, 12)( 3, 11)( 4, 18)( 5, 17)( 6, 16)( 7, 15)( 8, 14)
( 9, 13)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 37)( 29, 39)( 30, 38)
( 31, 45)( 32, 44)( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 47, 48)( 49, 54)
( 50, 53)( 51, 52)( 55, 91)( 56, 93)( 57, 92)( 58, 99)( 59, 98)( 60, 97)
( 61, 96)( 62, 95)( 63, 94)( 64, 82)( 65, 84)( 66, 83)( 67, 90)( 68, 89)
( 69, 88)( 70, 87)( 71, 86)( 72, 85)( 73,100)( 74,102)( 75,101)( 76,108)
( 77,107)( 78,106)( 79,105)( 80,104)( 81,103)(109,172)(110,174)(111,173)
(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)(118,163)(119,165)
(120,164)(121,171)(122,170)(123,169)(124,168)(125,167)(126,166)(127,181)
(128,183)(129,182)(130,189)(131,188)(132,187)(133,186)(134,185)(135,184)
(136,199)(137,201)(138,200)(139,207)(140,206)(141,205)(142,204)(143,203)
(144,202)(145,190)(146,192)(147,191)(148,198)(149,197)(150,196)(151,195)
(152,194)(153,193)(154,208)(155,210)(156,209)(157,216)(158,215)(159,214)
(160,213)(161,212)(162,211)(217,334)(218,336)(219,335)(220,342)(221,341)
(222,340)(223,339)(224,338)(225,337)(226,325)(227,327)(228,326)(229,333)
(230,332)(231,331)(232,330)(233,329)(234,328)(235,343)(236,345)(237,344)
(238,351)(239,350)(240,349)(241,348)(242,347)(243,346)(244,361)(245,363)
(246,362)(247,369)(248,368)(249,367)(250,366)(251,365)(252,364)(253,352)
(254,354)(255,353)(256,360)(257,359)(258,358)(259,357)(260,356)(261,355)
(262,370)(263,372)(264,371)(265,378)(266,377)(267,376)(268,375)(269,374)
(270,373)(271,415)(272,417)(273,416)(274,423)(275,422)(276,421)(277,420)
(278,419)(279,418)(280,406)(281,408)(282,407)(283,414)(284,413)(285,412)
(286,411)(287,410)(288,409)(289,424)(290,426)(291,425)(292,432)(293,431)
(294,430)(295,429)(296,428)(297,427)(298,388)(299,390)(300,389)(301,396)
(302,395)(303,394)(304,393)(305,392)(306,391)(307,379)(308,381)(309,380)
(310,387)(311,386)(312,385)(313,384)(314,383)(315,382)(316,397)(317,399)
(318,398)(319,405)(320,404)(321,403)(322,402)(323,401)(324,400);;
s2 := ( 1,220)( 2,222)( 3,221)( 4,217)( 5,219)( 6,218)( 7,225)( 8,224)
( 9,223)( 10,238)( 11,240)( 12,239)( 13,235)( 14,237)( 15,236)( 16,243)
( 17,242)( 18,241)( 19,229)( 20,231)( 21,230)( 22,226)( 23,228)( 24,227)
( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)( 32,246)
( 33,245)( 34,252)( 35,251)( 36,250)( 37,265)( 38,267)( 39,266)( 40,262)
( 41,264)( 42,263)( 43,270)( 44,269)( 45,268)( 46,256)( 47,258)( 48,257)
( 49,253)( 50,255)( 51,254)( 52,261)( 53,260)( 54,259)( 55,301)( 56,303)
( 57,302)( 58,298)( 59,300)( 60,299)( 61,306)( 62,305)( 63,304)( 64,319)
( 65,321)( 66,320)( 67,316)( 68,318)( 69,317)( 70,324)( 71,323)( 72,322)
( 73,310)( 74,312)( 75,311)( 76,307)( 77,309)( 78,308)( 79,315)( 80,314)
( 81,313)( 82,274)( 83,276)( 84,275)( 85,271)( 86,273)( 87,272)( 88,279)
( 89,278)( 90,277)( 91,292)( 92,294)( 93,293)( 94,289)( 95,291)( 96,290)
( 97,297)( 98,296)( 99,295)(100,283)(101,285)(102,284)(103,280)(104,282)
(105,281)(106,288)(107,287)(108,286)(109,382)(110,384)(111,383)(112,379)
(113,381)(114,380)(115,387)(116,386)(117,385)(118,400)(119,402)(120,401)
(121,397)(122,399)(123,398)(124,405)(125,404)(126,403)(127,391)(128,393)
(129,392)(130,388)(131,390)(132,389)(133,396)(134,395)(135,394)(136,409)
(137,411)(138,410)(139,406)(140,408)(141,407)(142,414)(143,413)(144,412)
(145,427)(146,429)(147,428)(148,424)(149,426)(150,425)(151,432)(152,431)
(153,430)(154,418)(155,420)(156,419)(157,415)(158,417)(159,416)(160,423)
(161,422)(162,421)(163,328)(164,330)(165,329)(166,325)(167,327)(168,326)
(169,333)(170,332)(171,331)(172,346)(173,348)(174,347)(175,343)(176,345)
(177,344)(178,351)(179,350)(180,349)(181,337)(182,339)(183,338)(184,334)
(185,336)(186,335)(187,342)(188,341)(189,340)(190,355)(191,357)(192,356)
(193,352)(194,354)(195,353)(196,360)(197,359)(198,358)(199,373)(200,375)
(201,374)(202,370)(203,372)(204,371)(205,378)(206,377)(207,376)(208,364)
(209,366)(210,365)(211,361)(212,363)(213,362)(214,369)(215,368)(216,367);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
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*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*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(432)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)
(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)
(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(172,181)
(173,182)(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189)
(199,208)(200,209)(201,210)(202,211)(203,212)(204,213)(205,214)(206,215)
(207,216)(226,235)(227,236)(228,237)(229,238)(230,239)(231,240)(232,241)
(233,242)(234,243)(253,262)(254,263)(255,264)(256,265)(257,266)(258,267)
(259,268)(260,269)(261,270)(280,289)(281,290)(282,291)(283,292)(284,293)
(285,294)(286,295)(287,296)(288,297)(307,316)(308,317)(309,318)(310,319)
(311,320)(312,321)(313,322)(314,323)(315,324)(334,343)(335,344)(336,345)
(337,346)(338,347)(339,348)(340,349)(341,350)(342,351)(361,370)(362,371)
(363,372)(364,373)(365,374)(366,375)(367,376)(368,377)(369,378)(388,397)
(389,398)(390,399)(391,400)(392,401)(393,402)(394,403)(395,404)(396,405)
(415,424)(416,425)(417,426)(418,427)(419,428)(420,429)(421,430)(422,431)
(423,432);
s1 := Sym(432)!( 1, 10)( 2, 12)( 3, 11)( 4, 18)( 5, 17)( 6, 16)( 7, 15)
( 8, 14)( 9, 13)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 37)( 29, 39)
( 30, 38)( 31, 45)( 32, 44)( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 47, 48)
( 49, 54)( 50, 53)( 51, 52)( 55, 91)( 56, 93)( 57, 92)( 58, 99)( 59, 98)
( 60, 97)( 61, 96)( 62, 95)( 63, 94)( 64, 82)( 65, 84)( 66, 83)( 67, 90)
( 68, 89)( 69, 88)( 70, 87)( 71, 86)( 72, 85)( 73,100)( 74,102)( 75,101)
( 76,108)( 77,107)( 78,106)( 79,105)( 80,104)( 81,103)(109,172)(110,174)
(111,173)(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)(118,163)
(119,165)(120,164)(121,171)(122,170)(123,169)(124,168)(125,167)(126,166)
(127,181)(128,183)(129,182)(130,189)(131,188)(132,187)(133,186)(134,185)
(135,184)(136,199)(137,201)(138,200)(139,207)(140,206)(141,205)(142,204)
(143,203)(144,202)(145,190)(146,192)(147,191)(148,198)(149,197)(150,196)
(151,195)(152,194)(153,193)(154,208)(155,210)(156,209)(157,216)(158,215)
(159,214)(160,213)(161,212)(162,211)(217,334)(218,336)(219,335)(220,342)
(221,341)(222,340)(223,339)(224,338)(225,337)(226,325)(227,327)(228,326)
(229,333)(230,332)(231,331)(232,330)(233,329)(234,328)(235,343)(236,345)
(237,344)(238,351)(239,350)(240,349)(241,348)(242,347)(243,346)(244,361)
(245,363)(246,362)(247,369)(248,368)(249,367)(250,366)(251,365)(252,364)
(253,352)(254,354)(255,353)(256,360)(257,359)(258,358)(259,357)(260,356)
(261,355)(262,370)(263,372)(264,371)(265,378)(266,377)(267,376)(268,375)
(269,374)(270,373)(271,415)(272,417)(273,416)(274,423)(275,422)(276,421)
(277,420)(278,419)(279,418)(280,406)(281,408)(282,407)(283,414)(284,413)
(285,412)(286,411)(287,410)(288,409)(289,424)(290,426)(291,425)(292,432)
(293,431)(294,430)(295,429)(296,428)(297,427)(298,388)(299,390)(300,389)
(301,396)(302,395)(303,394)(304,393)(305,392)(306,391)(307,379)(308,381)
(309,380)(310,387)(311,386)(312,385)(313,384)(314,383)(315,382)(316,397)
(317,399)(318,398)(319,405)(320,404)(321,403)(322,402)(323,401)(324,400);
s2 := Sym(432)!( 1,220)( 2,222)( 3,221)( 4,217)( 5,219)( 6,218)( 7,225)
( 8,224)( 9,223)( 10,238)( 11,240)( 12,239)( 13,235)( 14,237)( 15,236)
( 16,243)( 17,242)( 18,241)( 19,229)( 20,231)( 21,230)( 22,226)( 23,228)
( 24,227)( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)
( 32,246)( 33,245)( 34,252)( 35,251)( 36,250)( 37,265)( 38,267)( 39,266)
( 40,262)( 41,264)( 42,263)( 43,270)( 44,269)( 45,268)( 46,256)( 47,258)
( 48,257)( 49,253)( 50,255)( 51,254)( 52,261)( 53,260)( 54,259)( 55,301)
( 56,303)( 57,302)( 58,298)( 59,300)( 60,299)( 61,306)( 62,305)( 63,304)
( 64,319)( 65,321)( 66,320)( 67,316)( 68,318)( 69,317)( 70,324)( 71,323)
( 72,322)( 73,310)( 74,312)( 75,311)( 76,307)( 77,309)( 78,308)( 79,315)
( 80,314)( 81,313)( 82,274)( 83,276)( 84,275)( 85,271)( 86,273)( 87,272)
( 88,279)( 89,278)( 90,277)( 91,292)( 92,294)( 93,293)( 94,289)( 95,291)
( 96,290)( 97,297)( 98,296)( 99,295)(100,283)(101,285)(102,284)(103,280)
(104,282)(105,281)(106,288)(107,287)(108,286)(109,382)(110,384)(111,383)
(112,379)(113,381)(114,380)(115,387)(116,386)(117,385)(118,400)(119,402)
(120,401)(121,397)(122,399)(123,398)(124,405)(125,404)(126,403)(127,391)
(128,393)(129,392)(130,388)(131,390)(132,389)(133,396)(134,395)(135,394)
(136,409)(137,411)(138,410)(139,406)(140,408)(141,407)(142,414)(143,413)
(144,412)(145,427)(146,429)(147,428)(148,424)(149,426)(150,425)(151,432)
(152,431)(153,430)(154,418)(155,420)(156,419)(157,415)(158,417)(159,416)
(160,423)(161,422)(162,421)(163,328)(164,330)(165,329)(166,325)(167,327)
(168,326)(169,333)(170,332)(171,331)(172,346)(173,348)(174,347)(175,343)
(176,345)(177,344)(178,351)(179,350)(180,349)(181,337)(182,339)(183,338)
(184,334)(185,336)(186,335)(187,342)(188,341)(189,340)(190,355)(191,357)
(192,356)(193,352)(194,354)(195,353)(196,360)(197,359)(198,358)(199,373)
(200,375)(201,374)(202,370)(203,372)(204,371)(205,378)(206,377)(207,376)
(208,364)(209,366)(210,365)(211,361)(212,363)(213,362)(214,369)(215,368)
(216,367);
poly := sub<Sym(432)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
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*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*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 >;
References : None.
to this polytope