Polytope of Type {24,36}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,36}*1728a
if this polytope has a name.
Group : SmallGroup(1728,3359)
Rank : 3
Schlafli Type : {24,36}
Number of vertices, edges, etc : 24, 432, 36
Order of s0s1s2 : 72
Order of s0s1s2s1 : 12
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
3-fold quotients : {8,36}*576b, {24,12}*576a
4-fold quotients : {6,36}*432b, {12,18}*432b
6-fold quotients : {4,36}*288a, {12,12}*288b
8-fold quotients : {6,18}*216b
9-fold quotients : {8,12}*192b
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
24-fold quotients : {2,18}*72, {6,6}*72b
27-fold quotients : {8,4}*64b
36-fold quotients : {2,12}*48, {4,6}*48a
48-fold quotients : {2,9}*36, {6,3}*36
54-fold quotients : {4,4}*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)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108)(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)(163,190)
(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)
(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)
(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)(187,211)
(188,212)(189,213)(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)
(271,379)(272,380)(273,381)(274,385)(275,386)(276,387)(277,382)(278,383)
(279,384)(280,388)(281,389)(282,390)(283,394)(284,395)(285,396)(286,391)
(287,392)(288,393)(289,397)(290,398)(291,399)(292,403)(293,404)(294,405)
(295,400)(296,401)(297,402)(298,406)(299,407)(300,408)(301,412)(302,413)
(303,414)(304,409)(305,410)(306,411)(307,415)(308,416)(309,417)(310,421)
(311,422)(312,423)(313,418)(314,419)(315,420)(316,424)(317,425)(318,426)
(319,430)(320,431)(321,432)(322,427)(323,428)(324,429);;
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, 91)( 56, 93)( 57, 92)( 58, 97)( 59, 99)( 60, 98)
( 61, 94)( 62, 96)( 63, 95)( 64, 82)( 65, 84)( 66, 83)( 67, 88)( 68, 90)
( 69, 89)( 70, 85)( 71, 87)( 72, 86)( 73,102)( 74,101)( 75,100)( 76,108)
( 77,107)( 78,106)( 79,105)( 80,104)( 81,103)(109,145)(110,147)(111,146)
(112,151)(113,153)(114,152)(115,148)(116,150)(117,149)(118,136)(119,138)
(120,137)(121,142)(122,144)(123,143)(124,139)(125,141)(126,140)(127,156)
(128,155)(129,154)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157)
(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)(217,388)(218,390)(219,389)(220,394)(221,396)(222,395)
(223,391)(224,393)(225,392)(226,379)(227,381)(228,380)(229,385)(230,387)
(231,386)(232,382)(233,384)(234,383)(235,399)(236,398)(237,397)(238,405)
(239,404)(240,403)(241,402)(242,401)(243,400)(244,415)(245,417)(246,416)
(247,421)(248,423)(249,422)(250,418)(251,420)(252,419)(253,406)(254,408)
(255,407)(256,412)(257,414)(258,413)(259,409)(260,411)(261,410)(262,426)
(263,425)(264,424)(265,432)(266,431)(267,430)(268,429)(269,428)(270,427)
(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*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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 ];;
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)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)
( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(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)
(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)
(171,195)(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)
(179,203)(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)
(187,211)(188,212)(189,213)(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)(271,379)(272,380)(273,381)(274,385)(275,386)(276,387)(277,382)
(278,383)(279,384)(280,388)(281,389)(282,390)(283,394)(284,395)(285,396)
(286,391)(287,392)(288,393)(289,397)(290,398)(291,399)(292,403)(293,404)
(294,405)(295,400)(296,401)(297,402)(298,406)(299,407)(300,408)(301,412)
(302,413)(303,414)(304,409)(305,410)(306,411)(307,415)(308,416)(309,417)
(310,421)(311,422)(312,423)(313,418)(314,419)(315,420)(316,424)(317,425)
(318,426)(319,430)(320,431)(321,432)(322,427)(323,428)(324,429);
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, 91)( 56, 93)( 57, 92)( 58, 97)( 59, 99)
( 60, 98)( 61, 94)( 62, 96)( 63, 95)( 64, 82)( 65, 84)( 66, 83)( 67, 88)
( 68, 90)( 69, 89)( 70, 85)( 71, 87)( 72, 86)( 73,102)( 74,101)( 75,100)
( 76,108)( 77,107)( 78,106)( 79,105)( 80,104)( 81,103)(109,145)(110,147)
(111,146)(112,151)(113,153)(114,152)(115,148)(116,150)(117,149)(118,136)
(119,138)(120,137)(121,142)(122,144)(123,143)(124,139)(125,141)(126,140)
(127,156)(128,155)(129,154)(130,162)(131,161)(132,160)(133,159)(134,158)
(135,157)(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)(217,388)(218,390)(219,389)(220,394)(221,396)
(222,395)(223,391)(224,393)(225,392)(226,379)(227,381)(228,380)(229,385)
(230,387)(231,386)(232,382)(233,384)(234,383)(235,399)(236,398)(237,397)
(238,405)(239,404)(240,403)(241,402)(242,401)(243,400)(244,415)(245,417)
(246,416)(247,421)(248,423)(249,422)(250,418)(251,420)(252,419)(253,406)
(254,408)(255,407)(256,412)(257,414)(258,413)(259,409)(260,411)(261,410)
(262,426)(263,425)(264,424)(265,432)(266,431)(267,430)(268,429)(269,428)
(270,427)(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*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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 >;
References : None.
to this polytope