Polytope of Type {6,24}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,24}*1728e
if this polytope has a name.
Group : SmallGroup(1728,30298)
Rank : 3
Schlafli Type : {6,24}
Number of vertices, edges, etc : 36, 432, 144
Order of s0s1s2 : 12
Order of s0s1s2s1 : 24
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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}*864b
3-fold quotients : {6,24}*576e
4-fold quotients : {6,12}*432d
6-fold quotients : {6,12}*288a
8-fold quotients : {6,6}*216b
9-fold quotients : {6,8}*192b
12-fold quotients : {6,12}*144d
16-fold quotients : {6,6}*108
18-fold quotients : {3,8}*96, {6,4}*96
24-fold quotients : {6,6}*72a
36-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
72-fold quotients : {3,4}*24, {2,6}*24, {6,2}*24
144-fold quotients : {2,3}*12, {3,2}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 13, 23)
( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)( 34, 42)
( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)( 53, 55)
( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)( 63, 69)
( 64, 70)( 73,145)( 74,146)( 75,148)( 76,147)( 77,151)( 78,152)( 79,149)
( 80,150)( 81,161)( 82,162)( 83,164)( 84,163)( 85,167)( 86,168)( 87,165)
( 88,166)( 89,153)( 90,154)( 91,156)( 92,155)( 93,159)( 94,160)( 95,157)
( 96,158)( 97,169)( 98,170)( 99,172)(100,171)(101,175)(102,176)(103,173)
(104,174)(105,185)(106,186)(107,188)(108,187)(109,191)(110,192)(111,189)
(112,190)(113,177)(114,178)(115,180)(116,179)(117,183)(118,184)(119,181)
(120,182)(121,193)(122,194)(123,196)(124,195)(125,199)(126,200)(127,197)
(128,198)(129,209)(130,210)(131,212)(132,211)(133,215)(134,216)(135,213)
(136,214)(137,201)(138,202)(139,204)(140,203)(141,207)(142,208)(143,205)
(144,206)(217,218)(221,224)(222,223)(225,234)(226,233)(227,235)(228,236)
(229,240)(230,239)(231,238)(232,237)(241,242)(245,248)(246,247)(249,258)
(250,257)(251,259)(252,260)(253,264)(254,263)(255,262)(256,261)(265,266)
(269,272)(270,271)(273,282)(274,281)(275,283)(276,284)(277,288)(278,287)
(279,286)(280,285)(289,362)(290,361)(291,363)(292,364)(293,368)(294,367)
(295,366)(296,365)(297,378)(298,377)(299,379)(300,380)(301,384)(302,383)
(303,382)(304,381)(305,370)(306,369)(307,371)(308,372)(309,376)(310,375)
(311,374)(312,373)(313,386)(314,385)(315,387)(316,388)(317,392)(318,391)
(319,390)(320,389)(321,402)(322,401)(323,403)(324,404)(325,408)(326,407)
(327,406)(328,405)(329,394)(330,393)(331,395)(332,396)(333,400)(334,399)
(335,398)(336,397)(337,410)(338,409)(339,411)(340,412)(341,416)(342,415)
(343,414)(344,413)(345,426)(346,425)(347,427)(348,428)(349,432)(350,431)
(351,430)(352,429)(353,418)(354,417)(355,419)(356,420)(357,424)(358,423)
(359,422)(360,421);;
s1 := ( 1, 73)( 2, 74)( 3, 79)( 4, 80)( 5, 78)( 6, 77)( 7, 75)( 8, 76)
( 9, 81)( 10, 82)( 11, 87)( 12, 88)( 13, 86)( 14, 85)( 15, 83)( 16, 84)
( 17, 89)( 18, 90)( 19, 95)( 20, 96)( 21, 94)( 22, 93)( 23, 91)( 24, 92)
( 25,137)( 26,138)( 27,143)( 28,144)( 29,142)( 30,141)( 31,139)( 32,140)
( 33,121)( 34,122)( 35,127)( 36,128)( 37,126)( 38,125)( 39,123)( 40,124)
( 41,129)( 42,130)( 43,135)( 44,136)( 45,134)( 46,133)( 47,131)( 48,132)
( 49,105)( 50,106)( 51,111)( 52,112)( 53,110)( 54,109)( 55,107)( 56,108)
( 57,113)( 58,114)( 59,119)( 60,120)( 61,118)( 62,117)( 63,115)( 64,116)
( 65, 97)( 66, 98)( 67,103)( 68,104)( 69,102)( 70,101)( 71, 99)( 72,100)
(147,151)(148,152)(149,150)(155,159)(156,160)(157,158)(163,167)(164,168)
(165,166)(169,209)(170,210)(171,215)(172,216)(173,214)(174,213)(175,211)
(176,212)(177,193)(178,194)(179,199)(180,200)(181,198)(182,197)(183,195)
(184,196)(185,201)(186,202)(187,207)(188,208)(189,206)(190,205)(191,203)
(192,204)(217,290)(218,289)(219,296)(220,295)(221,293)(222,294)(223,292)
(224,291)(225,298)(226,297)(227,304)(228,303)(229,301)(230,302)(231,300)
(232,299)(233,306)(234,305)(235,312)(236,311)(237,309)(238,310)(239,308)
(240,307)(241,354)(242,353)(243,360)(244,359)(245,357)(246,358)(247,356)
(248,355)(249,338)(250,337)(251,344)(252,343)(253,341)(254,342)(255,340)
(256,339)(257,346)(258,345)(259,352)(260,351)(261,349)(262,350)(263,348)
(264,347)(265,322)(266,321)(267,328)(268,327)(269,325)(270,326)(271,324)
(272,323)(273,330)(274,329)(275,336)(276,335)(277,333)(278,334)(279,332)
(280,331)(281,314)(282,313)(283,320)(284,319)(285,317)(286,318)(287,316)
(288,315)(361,362)(363,368)(364,367)(369,370)(371,376)(372,375)(377,378)
(379,384)(380,383)(385,426)(386,425)(387,432)(388,431)(389,429)(390,430)
(391,428)(392,427)(393,410)(394,409)(395,416)(396,415)(397,413)(398,414)
(399,412)(400,411)(401,418)(402,417)(403,424)(404,423)(405,421)(406,422)
(407,420)(408,419);;
s2 := ( 1,243)( 2,244)( 3,242)( 4,241)( 5,247)( 6,248)( 7,246)( 8,245)
( 9,259)( 10,260)( 11,258)( 12,257)( 13,263)( 14,264)( 15,262)( 16,261)
( 17,251)( 18,252)( 19,250)( 20,249)( 21,255)( 22,256)( 23,254)( 24,253)
( 25,219)( 26,220)( 27,218)( 28,217)( 29,223)( 30,224)( 31,222)( 32,221)
( 33,235)( 34,236)( 35,234)( 36,233)( 37,239)( 38,240)( 39,238)( 40,237)
( 41,227)( 42,228)( 43,226)( 44,225)( 45,231)( 46,232)( 47,230)( 48,229)
( 49,267)( 50,268)( 51,266)( 52,265)( 53,271)( 54,272)( 55,270)( 56,269)
( 57,283)( 58,284)( 59,282)( 60,281)( 61,287)( 62,288)( 63,286)( 64,285)
( 65,275)( 66,276)( 67,274)( 68,273)( 69,279)( 70,280)( 71,278)( 72,277)
( 73,315)( 74,316)( 75,314)( 76,313)( 77,319)( 78,320)( 79,318)( 80,317)
( 81,331)( 82,332)( 83,330)( 84,329)( 85,335)( 86,336)( 87,334)( 88,333)
( 89,323)( 90,324)( 91,322)( 92,321)( 93,327)( 94,328)( 95,326)( 96,325)
( 97,291)( 98,292)( 99,290)(100,289)(101,295)(102,296)(103,294)(104,293)
(105,307)(106,308)(107,306)(108,305)(109,311)(110,312)(111,310)(112,309)
(113,299)(114,300)(115,298)(116,297)(117,303)(118,304)(119,302)(120,301)
(121,339)(122,340)(123,338)(124,337)(125,343)(126,344)(127,342)(128,341)
(129,355)(130,356)(131,354)(132,353)(133,359)(134,360)(135,358)(136,357)
(137,347)(138,348)(139,346)(140,345)(141,351)(142,352)(143,350)(144,349)
(145,387)(146,388)(147,386)(148,385)(149,391)(150,392)(151,390)(152,389)
(153,403)(154,404)(155,402)(156,401)(157,407)(158,408)(159,406)(160,405)
(161,395)(162,396)(163,394)(164,393)(165,399)(166,400)(167,398)(168,397)
(169,363)(170,364)(171,362)(172,361)(173,367)(174,368)(175,366)(176,365)
(177,379)(178,380)(179,378)(180,377)(181,383)(182,384)(183,382)(184,381)
(185,371)(186,372)(187,370)(188,369)(189,375)(190,376)(191,374)(192,373)
(193,411)(194,412)(195,410)(196,409)(197,415)(198,416)(199,414)(200,413)
(201,427)(202,428)(203,426)(204,425)(205,431)(206,432)(207,430)(208,429)
(209,419)(210,420)(211,418)(212,417)(213,423)(214,424)(215,422)(216,421);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)
( 13, 23)( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)
( 34, 42)( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)
( 53, 55)( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)
( 63, 69)( 64, 70)( 73,145)( 74,146)( 75,148)( 76,147)( 77,151)( 78,152)
( 79,149)( 80,150)( 81,161)( 82,162)( 83,164)( 84,163)( 85,167)( 86,168)
( 87,165)( 88,166)( 89,153)( 90,154)( 91,156)( 92,155)( 93,159)( 94,160)
( 95,157)( 96,158)( 97,169)( 98,170)( 99,172)(100,171)(101,175)(102,176)
(103,173)(104,174)(105,185)(106,186)(107,188)(108,187)(109,191)(110,192)
(111,189)(112,190)(113,177)(114,178)(115,180)(116,179)(117,183)(118,184)
(119,181)(120,182)(121,193)(122,194)(123,196)(124,195)(125,199)(126,200)
(127,197)(128,198)(129,209)(130,210)(131,212)(132,211)(133,215)(134,216)
(135,213)(136,214)(137,201)(138,202)(139,204)(140,203)(141,207)(142,208)
(143,205)(144,206)(217,218)(221,224)(222,223)(225,234)(226,233)(227,235)
(228,236)(229,240)(230,239)(231,238)(232,237)(241,242)(245,248)(246,247)
(249,258)(250,257)(251,259)(252,260)(253,264)(254,263)(255,262)(256,261)
(265,266)(269,272)(270,271)(273,282)(274,281)(275,283)(276,284)(277,288)
(278,287)(279,286)(280,285)(289,362)(290,361)(291,363)(292,364)(293,368)
(294,367)(295,366)(296,365)(297,378)(298,377)(299,379)(300,380)(301,384)
(302,383)(303,382)(304,381)(305,370)(306,369)(307,371)(308,372)(309,376)
(310,375)(311,374)(312,373)(313,386)(314,385)(315,387)(316,388)(317,392)
(318,391)(319,390)(320,389)(321,402)(322,401)(323,403)(324,404)(325,408)
(326,407)(327,406)(328,405)(329,394)(330,393)(331,395)(332,396)(333,400)
(334,399)(335,398)(336,397)(337,410)(338,409)(339,411)(340,412)(341,416)
(342,415)(343,414)(344,413)(345,426)(346,425)(347,427)(348,428)(349,432)
(350,431)(351,430)(352,429)(353,418)(354,417)(355,419)(356,420)(357,424)
(358,423)(359,422)(360,421);
s1 := Sym(432)!( 1, 73)( 2, 74)( 3, 79)( 4, 80)( 5, 78)( 6, 77)( 7, 75)
( 8, 76)( 9, 81)( 10, 82)( 11, 87)( 12, 88)( 13, 86)( 14, 85)( 15, 83)
( 16, 84)( 17, 89)( 18, 90)( 19, 95)( 20, 96)( 21, 94)( 22, 93)( 23, 91)
( 24, 92)( 25,137)( 26,138)( 27,143)( 28,144)( 29,142)( 30,141)( 31,139)
( 32,140)( 33,121)( 34,122)( 35,127)( 36,128)( 37,126)( 38,125)( 39,123)
( 40,124)( 41,129)( 42,130)( 43,135)( 44,136)( 45,134)( 46,133)( 47,131)
( 48,132)( 49,105)( 50,106)( 51,111)( 52,112)( 53,110)( 54,109)( 55,107)
( 56,108)( 57,113)( 58,114)( 59,119)( 60,120)( 61,118)( 62,117)( 63,115)
( 64,116)( 65, 97)( 66, 98)( 67,103)( 68,104)( 69,102)( 70,101)( 71, 99)
( 72,100)(147,151)(148,152)(149,150)(155,159)(156,160)(157,158)(163,167)
(164,168)(165,166)(169,209)(170,210)(171,215)(172,216)(173,214)(174,213)
(175,211)(176,212)(177,193)(178,194)(179,199)(180,200)(181,198)(182,197)
(183,195)(184,196)(185,201)(186,202)(187,207)(188,208)(189,206)(190,205)
(191,203)(192,204)(217,290)(218,289)(219,296)(220,295)(221,293)(222,294)
(223,292)(224,291)(225,298)(226,297)(227,304)(228,303)(229,301)(230,302)
(231,300)(232,299)(233,306)(234,305)(235,312)(236,311)(237,309)(238,310)
(239,308)(240,307)(241,354)(242,353)(243,360)(244,359)(245,357)(246,358)
(247,356)(248,355)(249,338)(250,337)(251,344)(252,343)(253,341)(254,342)
(255,340)(256,339)(257,346)(258,345)(259,352)(260,351)(261,349)(262,350)
(263,348)(264,347)(265,322)(266,321)(267,328)(268,327)(269,325)(270,326)
(271,324)(272,323)(273,330)(274,329)(275,336)(276,335)(277,333)(278,334)
(279,332)(280,331)(281,314)(282,313)(283,320)(284,319)(285,317)(286,318)
(287,316)(288,315)(361,362)(363,368)(364,367)(369,370)(371,376)(372,375)
(377,378)(379,384)(380,383)(385,426)(386,425)(387,432)(388,431)(389,429)
(390,430)(391,428)(392,427)(393,410)(394,409)(395,416)(396,415)(397,413)
(398,414)(399,412)(400,411)(401,418)(402,417)(403,424)(404,423)(405,421)
(406,422)(407,420)(408,419);
s2 := Sym(432)!( 1,243)( 2,244)( 3,242)( 4,241)( 5,247)( 6,248)( 7,246)
( 8,245)( 9,259)( 10,260)( 11,258)( 12,257)( 13,263)( 14,264)( 15,262)
( 16,261)( 17,251)( 18,252)( 19,250)( 20,249)( 21,255)( 22,256)( 23,254)
( 24,253)( 25,219)( 26,220)( 27,218)( 28,217)( 29,223)( 30,224)( 31,222)
( 32,221)( 33,235)( 34,236)( 35,234)( 36,233)( 37,239)( 38,240)( 39,238)
( 40,237)( 41,227)( 42,228)( 43,226)( 44,225)( 45,231)( 46,232)( 47,230)
( 48,229)( 49,267)( 50,268)( 51,266)( 52,265)( 53,271)( 54,272)( 55,270)
( 56,269)( 57,283)( 58,284)( 59,282)( 60,281)( 61,287)( 62,288)( 63,286)
( 64,285)( 65,275)( 66,276)( 67,274)( 68,273)( 69,279)( 70,280)( 71,278)
( 72,277)( 73,315)( 74,316)( 75,314)( 76,313)( 77,319)( 78,320)( 79,318)
( 80,317)( 81,331)( 82,332)( 83,330)( 84,329)( 85,335)( 86,336)( 87,334)
( 88,333)( 89,323)( 90,324)( 91,322)( 92,321)( 93,327)( 94,328)( 95,326)
( 96,325)( 97,291)( 98,292)( 99,290)(100,289)(101,295)(102,296)(103,294)
(104,293)(105,307)(106,308)(107,306)(108,305)(109,311)(110,312)(111,310)
(112,309)(113,299)(114,300)(115,298)(116,297)(117,303)(118,304)(119,302)
(120,301)(121,339)(122,340)(123,338)(124,337)(125,343)(126,344)(127,342)
(128,341)(129,355)(130,356)(131,354)(132,353)(133,359)(134,360)(135,358)
(136,357)(137,347)(138,348)(139,346)(140,345)(141,351)(142,352)(143,350)
(144,349)(145,387)(146,388)(147,386)(148,385)(149,391)(150,392)(151,390)
(152,389)(153,403)(154,404)(155,402)(156,401)(157,407)(158,408)(159,406)
(160,405)(161,395)(162,396)(163,394)(164,393)(165,399)(166,400)(167,398)
(168,397)(169,363)(170,364)(171,362)(172,361)(173,367)(174,368)(175,366)
(176,365)(177,379)(178,380)(179,378)(180,377)(181,383)(182,384)(183,382)
(184,381)(185,371)(186,372)(187,370)(188,369)(189,375)(190,376)(191,374)
(192,373)(193,411)(194,412)(195,410)(196,409)(197,415)(198,416)(199,414)
(200,413)(201,427)(202,428)(203,426)(204,425)(205,431)(206,432)(207,430)
(208,429)(209,419)(210,420)(211,418)(212,417)(213,423)(214,424)(215,422)
(216,421);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1 >;
References : None.
to this polytope