Polytope of Type {24,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,36}*1728b
if this polytope has a name.
Group : SmallGroup(1728,3898)
Rank : 3
Schlafli Type : {24,36}
Number of vertices, edges, etc : 24, 432, 36
Order of s₀s₁s₂ : 72
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 : {12,36}*864b, {24,18}*864b
   3-fold quotients : {8,36}*576a, {24,12}*576b
   4-fold quotients : {6,36}*432b, {12,18}*432b
   6-fold quotients : {4,36}*288a, {8,18}*288, {12,12}*288b, {24,6}*288c
   8-fold quotients : {6,18}*216b
   9-fold quotients : {8,12}*192a
   12-fold quotients : {2,36}*144, {4,18}*144a, {6,12}*144b, {12,6}*144c
   16-fold quotients : {6,9}*108
   18-fold quotients : {4,12}*96a, {8,6}*96
   24-fold quotients : {2,18}*72, {6,6}*72b
   27-fold quotients : {8,4}*64a
   36-fold quotients : {2,12}*48, {4,6}*48a
   48-fold quotients : {2,9}*36, {6,3}*36
   54-fold quotients : {4,4}*32, {8,2}*32
   72-fold quotients : {2,6}*24
   108-fold quotients : {2,4}*16, {4,2}*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 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)
( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)( 68, 98)
( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)( 76,106)
( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(109,136)(110,137)(111,138)
(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)
(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)
(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)
(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)
(186,189)(193,196)(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)
(212,215)(213,216)(217,325)(218,326)(219,327)(220,331)(221,332)(222,333)
(223,328)(224,329)(225,330)(226,334)(227,335)(228,336)(229,340)(230,341)
(231,342)(232,337)(233,338)(234,339)(235,343)(236,344)(237,345)(238,349)
(239,350)(240,351)(241,346)(242,347)(243,348)(244,352)(245,353)(246,354)
(247,358)(248,359)(249,360)(250,355)(251,356)(252,357)(253,361)(254,362)
(255,363)(256,367)(257,368)(258,369)(259,364)(260,365)(261,366)(262,370)
(263,371)(264,372)(265,376)(266,377)(267,378)(268,373)(269,374)(270,375)
(271,406)(272,407)(273,408)(274,412)(275,413)(276,414)(277,409)(278,410)
(279,411)(280,415)(281,416)(282,417)(283,421)(284,422)(285,423)(286,418)
(287,419)(288,420)(289,424)(290,425)(291,426)(292,430)(293,431)(294,432)
(295,427)(296,428)(297,429)(298,379)(299,380)(300,381)(301,385)(302,386)
(303,387)(304,382)(305,383)(306,384)(307,388)(308,389)(309,390)(310,394)
(311,395)(312,396)(313,391)(314,392)(315,393)(316,397)(317,398)(318,399)
(319,403)(320,404)(321,405)(322,400)(323,401)(324,402);;
s1 := (  1,220)(  2,222)(  3,221)(  4,217)(  5,219)(  6,218)(  7,223)(  8,225)
(  9,224)( 10,240)( 11,239)( 12,238)( 13,237)( 14,236)( 15,235)( 16,243)
( 17,242)( 18,241)( 19,231)( 20,230)( 21,229)( 22,228)( 23,227)( 24,226)
( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)( 32,246)
( 33,245)( 34,250)( 35,252)( 36,251)( 37,267)( 38,266)( 39,265)( 40,264)
( 41,263)( 42,262)( 43,270)( 44,269)( 45,268)( 46,258)( 47,257)( 48,256)
( 49,255)( 50,254)( 51,253)( 52,261)( 53,260)( 54,259)( 55,301)( 56,303)
( 57,302)( 58,298)( 59,300)( 60,299)( 61,304)( 62,306)( 63,305)( 64,321)
( 65,320)( 66,319)( 67,318)( 68,317)( 69,316)( 70,324)( 71,323)( 72,322)
( 73,312)( 74,311)( 75,310)( 76,309)( 77,308)( 78,307)( 79,315)( 80,314)
( 81,313)( 82,274)( 83,276)( 84,275)( 85,271)( 86,273)( 87,272)( 88,277)
( 89,279)( 90,278)( 91,294)( 92,293)( 93,292)( 94,291)( 95,290)( 96,289)
( 97,297)( 98,296)( 99,295)(100,285)(101,284)(102,283)(103,282)(104,281)
(105,280)(106,288)(107,287)(108,286)(109,355)(110,357)(111,356)(112,352)
(113,354)(114,353)(115,358)(116,360)(117,359)(118,375)(119,374)(120,373)
(121,372)(122,371)(123,370)(124,378)(125,377)(126,376)(127,366)(128,365)
(129,364)(130,363)(131,362)(132,361)(133,369)(134,368)(135,367)(136,328)
(137,330)(138,329)(139,325)(140,327)(141,326)(142,331)(143,333)(144,332)
(145,348)(146,347)(147,346)(148,345)(149,344)(150,343)(151,351)(152,350)
(153,349)(154,339)(155,338)(156,337)(157,336)(158,335)(159,334)(160,342)
(161,341)(162,340)(163,382)(164,384)(165,383)(166,379)(167,381)(168,380)
(169,385)(170,387)(171,386)(172,402)(173,401)(174,400)(175,399)(176,398)
(177,397)(178,405)(179,404)(180,403)(181,393)(182,392)(183,391)(184,390)
(185,389)(186,388)(187,396)(188,395)(189,394)(190,409)(191,411)(192,410)
(193,406)(194,408)(195,407)(196,412)(197,414)(198,413)(199,429)(200,428)
(201,427)(202,426)(203,425)(204,424)(205,432)(206,431)(207,430)(208,420)
(209,419)(210,418)(211,417)(212,416)(213,415)(214,423)(215,422)(216,421);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)
(  9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 37)( 29, 39)( 30, 38)
( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 48)( 49, 54)
( 50, 53)( 51, 52)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)
( 61, 67)( 62, 69)( 63, 68)( 73, 75)( 76, 81)( 77, 80)( 78, 79)( 82, 91)
( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)( 90, 95)
(100,102)(103,108)(104,107)(105,106)(109,118)(110,120)(111,119)(112,124)
(113,126)(114,125)(115,121)(116,123)(117,122)(127,129)(130,135)(131,134)
(132,133)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)(142,148)
(143,150)(144,149)(154,156)(157,162)(158,161)(159,160)(163,172)(164,174)
(165,173)(166,178)(167,180)(168,179)(169,175)(170,177)(171,176)(181,183)
(184,189)(185,188)(186,187)(190,199)(191,201)(192,200)(193,205)(194,207)
(195,206)(196,202)(197,204)(198,203)(208,210)(211,216)(212,215)(213,214)
(217,415)(218,417)(219,416)(220,421)(221,423)(222,422)(223,418)(224,420)
(225,419)(226,406)(227,408)(228,407)(229,412)(230,414)(231,413)(232,409)
(233,411)(234,410)(235,426)(236,425)(237,424)(238,432)(239,431)(240,430)
(241,429)(242,428)(243,427)(244,388)(245,390)(246,389)(247,394)(248,396)
(249,395)(250,391)(251,393)(252,392)(253,379)(254,381)(255,380)(256,385)
(257,387)(258,386)(259,382)(260,384)(261,383)(262,399)(263,398)(264,397)
(265,405)(266,404)(267,403)(268,402)(269,401)(270,400)(271,334)(272,336)
(273,335)(274,340)(275,342)(276,341)(277,337)(278,339)(279,338)(280,325)
(281,327)(282,326)(283,331)(284,333)(285,332)(286,328)(287,330)(288,329)
(289,345)(290,344)(291,343)(292,351)(293,350)(294,349)(295,348)(296,347)
(297,346)(298,361)(299,363)(300,362)(301,367)(302,369)(303,368)(304,364)
(305,366)(306,365)(307,352)(308,354)(309,353)(310,358)(311,360)(312,359)
(313,355)(314,357)(315,356)(316,372)(317,371)(318,370)(319,378)(320,377)
(321,376)(322,375)(323,374)(324,373);;
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*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(432)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)
( 60, 90)( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)
( 68, 98)( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)
( 76,106)( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(109,136)(110,137)
(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)
(119,146)(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)
(127,154)(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)
(135,159)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)
(185,188)(186,189)(193,196)(194,197)(195,198)(202,205)(203,206)(204,207)
(211,214)(212,215)(213,216)(217,325)(218,326)(219,327)(220,331)(221,332)
(222,333)(223,328)(224,329)(225,330)(226,334)(227,335)(228,336)(229,340)
(230,341)(231,342)(232,337)(233,338)(234,339)(235,343)(236,344)(237,345)
(238,349)(239,350)(240,351)(241,346)(242,347)(243,348)(244,352)(245,353)
(246,354)(247,358)(248,359)(249,360)(250,355)(251,356)(252,357)(253,361)
(254,362)(255,363)(256,367)(257,368)(258,369)(259,364)(260,365)(261,366)
(262,370)(263,371)(264,372)(265,376)(266,377)(267,378)(268,373)(269,374)
(270,375)(271,406)(272,407)(273,408)(274,412)(275,413)(276,414)(277,409)
(278,410)(279,411)(280,415)(281,416)(282,417)(283,421)(284,422)(285,423)
(286,418)(287,419)(288,420)(289,424)(290,425)(291,426)(292,430)(293,431)
(294,432)(295,427)(296,428)(297,429)(298,379)(299,380)(300,381)(301,385)
(302,386)(303,387)(304,382)(305,383)(306,384)(307,388)(308,389)(309,390)
(310,394)(311,395)(312,396)(313,391)(314,392)(315,393)(316,397)(317,398)
(318,399)(319,403)(320,404)(321,405)(322,400)(323,401)(324,402);
s1 := Sym(432)!(  1,220)(  2,222)(  3,221)(  4,217)(  5,219)(  6,218)(  7,223)
(  8,225)(  9,224)( 10,240)( 11,239)( 12,238)( 13,237)( 14,236)( 15,235)
( 16,243)( 17,242)( 18,241)( 19,231)( 20,230)( 21,229)( 22,228)( 23,227)
( 24,226)( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)
( 32,246)( 33,245)( 34,250)( 35,252)( 36,251)( 37,267)( 38,266)( 39,265)
( 40,264)( 41,263)( 42,262)( 43,270)( 44,269)( 45,268)( 46,258)( 47,257)
( 48,256)( 49,255)( 50,254)( 51,253)( 52,261)( 53,260)( 54,259)( 55,301)
( 56,303)( 57,302)( 58,298)( 59,300)( 60,299)( 61,304)( 62,306)( 63,305)
( 64,321)( 65,320)( 66,319)( 67,318)( 68,317)( 69,316)( 70,324)( 71,323)
( 72,322)( 73,312)( 74,311)( 75,310)( 76,309)( 77,308)( 78,307)( 79,315)
( 80,314)( 81,313)( 82,274)( 83,276)( 84,275)( 85,271)( 86,273)( 87,272)
( 88,277)( 89,279)( 90,278)( 91,294)( 92,293)( 93,292)( 94,291)( 95,290)
( 96,289)( 97,297)( 98,296)( 99,295)(100,285)(101,284)(102,283)(103,282)
(104,281)(105,280)(106,288)(107,287)(108,286)(109,355)(110,357)(111,356)
(112,352)(113,354)(114,353)(115,358)(116,360)(117,359)(118,375)(119,374)
(120,373)(121,372)(122,371)(123,370)(124,378)(125,377)(126,376)(127,366)
(128,365)(129,364)(130,363)(131,362)(132,361)(133,369)(134,368)(135,367)
(136,328)(137,330)(138,329)(139,325)(140,327)(141,326)(142,331)(143,333)
(144,332)(145,348)(146,347)(147,346)(148,345)(149,344)(150,343)(151,351)
(152,350)(153,349)(154,339)(155,338)(156,337)(157,336)(158,335)(159,334)
(160,342)(161,341)(162,340)(163,382)(164,384)(165,383)(166,379)(167,381)
(168,380)(169,385)(170,387)(171,386)(172,402)(173,401)(174,400)(175,399)
(176,398)(177,397)(178,405)(179,404)(180,403)(181,393)(182,392)(183,391)
(184,390)(185,389)(186,388)(187,396)(188,395)(189,394)(190,409)(191,411)
(192,410)(193,406)(194,408)(195,407)(196,412)(197,414)(198,413)(199,429)
(200,428)(201,427)(202,426)(203,425)(204,424)(205,432)(206,431)(207,430)
(208,420)(209,419)(210,418)(211,417)(212,416)(213,415)(214,423)(215,422)
(216,421);
s2 := Sym(432)!(  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)
(  8, 15)(  9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 37)( 29, 39)
( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 48)
( 49, 54)( 50, 53)( 51, 52)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)
( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 73, 75)( 76, 81)( 77, 80)( 78, 79)
( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)
( 90, 95)(100,102)(103,108)(104,107)(105,106)(109,118)(110,120)(111,119)
(112,124)(113,126)(114,125)(115,121)(116,123)(117,122)(127,129)(130,135)
(131,134)(132,133)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)
(142,148)(143,150)(144,149)(154,156)(157,162)(158,161)(159,160)(163,172)
(164,174)(165,173)(166,178)(167,180)(168,179)(169,175)(170,177)(171,176)
(181,183)(184,189)(185,188)(186,187)(190,199)(191,201)(192,200)(193,205)
(194,207)(195,206)(196,202)(197,204)(198,203)(208,210)(211,216)(212,215)
(213,214)(217,415)(218,417)(219,416)(220,421)(221,423)(222,422)(223,418)
(224,420)(225,419)(226,406)(227,408)(228,407)(229,412)(230,414)(231,413)
(232,409)(233,411)(234,410)(235,426)(236,425)(237,424)(238,432)(239,431)
(240,430)(241,429)(242,428)(243,427)(244,388)(245,390)(246,389)(247,394)
(248,396)(249,395)(250,391)(251,393)(252,392)(253,379)(254,381)(255,380)
(256,385)(257,387)(258,386)(259,382)(260,384)(261,383)(262,399)(263,398)
(264,397)(265,405)(266,404)(267,403)(268,402)(269,401)(270,400)(271,334)
(272,336)(273,335)(274,340)(275,342)(276,341)(277,337)(278,339)(279,338)
(280,325)(281,327)(282,326)(283,331)(284,333)(285,332)(286,328)(287,330)
(288,329)(289,345)(290,344)(291,343)(292,351)(293,350)(294,349)(295,348)
(296,347)(297,346)(298,361)(299,363)(300,362)(301,367)(302,369)(303,368)
(304,364)(305,366)(306,365)(307,352)(308,354)(309,353)(310,358)(311,360)
(312,359)(313,355)(314,357)(315,356)(316,372)(317,371)(318,370)(319,378)
(320,377)(321,376)(322,375)(323,374)(324,373);
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*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*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 >;

References : None.
to this polytope