Polytope of Type {36,24}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,24}*1728b
if this polytope has a name.
Group : SmallGroup(1728,3898)
Rank : 3
Schlafli Type : {36,24}
Number of vertices, edges, etc : 36, 432, 24
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 : {36,12}*864b, {18,24}*864b
3-fold quotients : {36,8}*576a, {12,24}*576b
4-fold quotients : {36,6}*432b, {18,12}*432b
6-fold quotients : {36,4}*288a, {18,8}*288, {12,12}*288c, {6,24}*288c
8-fold quotients : {18,6}*216b
9-fold quotients : {12,8}*192a
12-fold quotients : {36,2}*144, {18,4}*144a, {12,6}*144b, {6,12}*144c
16-fold quotients : {9,6}*108
18-fold quotients : {12,4}*96a, {6,8}*96
24-fold quotients : {18,2}*72, {6,6}*72c
27-fold quotients : {4,8}*64a
36-fold quotients : {12,2}*48, {6,4}*48a
48-fold quotients : {9,2}*36, {3,6}*36
54-fold quotients : {4,4}*32, {2,8}*32
72-fold quotients : {6,2}*24
108-fold quotients : {2,4}*16, {4,2}*16
144-fold quotients : {3,2}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 10, 21)( 11, 20)( 12, 19)( 13, 27)
( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 48)( 38, 47)( 39, 46)( 40, 54)( 41, 53)( 42, 52)( 43, 51)
( 44, 50)( 45, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 75)( 65, 74)
( 66, 73)( 67, 81)( 68, 80)( 69, 79)( 70, 78)( 71, 77)( 72, 76)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 91,102)( 92,101)( 93,100)( 94,108)( 95,107)
( 96,106)( 97,105)( 98,104)( 99,103)(110,111)(112,115)(113,117)(114,116)
(118,129)(119,128)(120,127)(121,135)(122,134)(123,133)(124,132)(125,131)
(126,130)(137,138)(139,142)(140,144)(141,143)(145,156)(146,155)(147,154)
(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(164,165)(166,169)
(167,171)(168,170)(172,183)(173,182)(174,181)(175,189)(176,188)(177,187)
(178,186)(179,185)(180,184)(191,192)(193,196)(194,198)(195,197)(199,210)
(200,209)(201,208)(202,216)(203,215)(204,214)(205,213)(206,212)(207,211)
(217,406)(218,408)(219,407)(220,412)(221,414)(222,413)(223,409)(224,411)
(225,410)(226,426)(227,425)(228,424)(229,432)(230,431)(231,430)(232,429)
(233,428)(234,427)(235,417)(236,416)(237,415)(238,423)(239,422)(240,421)
(241,420)(242,419)(243,418)(244,379)(245,381)(246,380)(247,385)(248,387)
(249,386)(250,382)(251,384)(252,383)(253,399)(254,398)(255,397)(256,405)
(257,404)(258,403)(259,402)(260,401)(261,400)(262,390)(263,389)(264,388)
(265,396)(266,395)(267,394)(268,393)(269,392)(270,391)(271,325)(272,327)
(273,326)(274,331)(275,333)(276,332)(277,328)(278,330)(279,329)(280,345)
(281,344)(282,343)(283,351)(284,350)(285,349)(286,348)(287,347)(288,346)
(289,336)(290,335)(291,334)(292,342)(293,341)(294,340)(295,339)(296,338)
(297,337)(298,352)(299,354)(300,353)(301,358)(302,360)(303,359)(304,355)
(305,357)(306,356)(307,372)(308,371)(309,370)(310,378)(311,377)(312,376)
(313,375)(314,374)(315,373)(316,363)(317,362)(318,361)(319,369)(320,368)
(321,367)(322,366)(323,365)(324,364);;
s1 := ( 1,229)( 2,231)( 3,230)( 4,226)( 5,228)( 6,227)( 7,232)( 8,234)
( 9,233)( 10,220)( 11,222)( 12,221)( 13,217)( 14,219)( 15,218)( 16,223)
( 17,225)( 18,224)( 19,240)( 20,239)( 21,238)( 22,237)( 23,236)( 24,235)
( 25,243)( 26,242)( 27,241)( 28,256)( 29,258)( 30,257)( 31,253)( 32,255)
( 33,254)( 34,259)( 35,261)( 36,260)( 37,247)( 38,249)( 39,248)( 40,244)
( 41,246)( 42,245)( 43,250)( 44,252)( 45,251)( 46,267)( 47,266)( 48,265)
( 49,264)( 50,263)( 51,262)( 52,270)( 53,269)( 54,268)( 55,310)( 56,312)
( 57,311)( 58,307)( 59,309)( 60,308)( 61,313)( 62,315)( 63,314)( 64,301)
( 65,303)( 66,302)( 67,298)( 68,300)( 69,299)( 70,304)( 71,306)( 72,305)
( 73,321)( 74,320)( 75,319)( 76,318)( 77,317)( 78,316)( 79,324)( 80,323)
( 81,322)( 82,283)( 83,285)( 84,284)( 85,280)( 86,282)( 87,281)( 88,286)
( 89,288)( 90,287)( 91,274)( 92,276)( 93,275)( 94,271)( 95,273)( 96,272)
( 97,277)( 98,279)( 99,278)(100,294)(101,293)(102,292)(103,291)(104,290)
(105,289)(106,297)(107,296)(108,295)(109,364)(110,366)(111,365)(112,361)
(113,363)(114,362)(115,367)(116,369)(117,368)(118,355)(119,357)(120,356)
(121,352)(122,354)(123,353)(124,358)(125,360)(126,359)(127,375)(128,374)
(129,373)(130,372)(131,371)(132,370)(133,378)(134,377)(135,376)(136,337)
(137,339)(138,338)(139,334)(140,336)(141,335)(142,340)(143,342)(144,341)
(145,328)(146,330)(147,329)(148,325)(149,327)(150,326)(151,331)(152,333)
(153,332)(154,348)(155,347)(156,346)(157,345)(158,344)(159,343)(160,351)
(161,350)(162,349)(163,391)(164,393)(165,392)(166,388)(167,390)(168,389)
(169,394)(170,396)(171,395)(172,382)(173,384)(174,383)(175,379)(176,381)
(177,380)(178,385)(179,387)(180,386)(181,402)(182,401)(183,400)(184,399)
(185,398)(186,397)(187,405)(188,404)(189,403)(190,418)(191,420)(192,419)
(193,415)(194,417)(195,416)(196,421)(197,423)(198,422)(199,409)(200,411)
(201,410)(202,406)(203,408)(204,407)(205,412)(206,414)(207,413)(208,429)
(209,428)(210,427)(211,426)(212,425)(213,424)(214,432)(215,431)(216,430);;
s2 := ( 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);;
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*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*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*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 10, 21)( 11, 20)( 12, 19)
( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 34)
( 32, 36)( 33, 35)( 37, 48)( 38, 47)( 39, 46)( 40, 54)( 41, 53)( 42, 52)
( 43, 51)( 44, 50)( 45, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 75)
( 65, 74)( 66, 73)( 67, 81)( 68, 80)( 69, 79)( 70, 78)( 71, 77)( 72, 76)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 91,102)( 92,101)( 93,100)( 94,108)
( 95,107)( 96,106)( 97,105)( 98,104)( 99,103)(110,111)(112,115)(113,117)
(114,116)(118,129)(119,128)(120,127)(121,135)(122,134)(123,133)(124,132)
(125,131)(126,130)(137,138)(139,142)(140,144)(141,143)(145,156)(146,155)
(147,154)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(164,165)
(166,169)(167,171)(168,170)(172,183)(173,182)(174,181)(175,189)(176,188)
(177,187)(178,186)(179,185)(180,184)(191,192)(193,196)(194,198)(195,197)
(199,210)(200,209)(201,208)(202,216)(203,215)(204,214)(205,213)(206,212)
(207,211)(217,406)(218,408)(219,407)(220,412)(221,414)(222,413)(223,409)
(224,411)(225,410)(226,426)(227,425)(228,424)(229,432)(230,431)(231,430)
(232,429)(233,428)(234,427)(235,417)(236,416)(237,415)(238,423)(239,422)
(240,421)(241,420)(242,419)(243,418)(244,379)(245,381)(246,380)(247,385)
(248,387)(249,386)(250,382)(251,384)(252,383)(253,399)(254,398)(255,397)
(256,405)(257,404)(258,403)(259,402)(260,401)(261,400)(262,390)(263,389)
(264,388)(265,396)(266,395)(267,394)(268,393)(269,392)(270,391)(271,325)
(272,327)(273,326)(274,331)(275,333)(276,332)(277,328)(278,330)(279,329)
(280,345)(281,344)(282,343)(283,351)(284,350)(285,349)(286,348)(287,347)
(288,346)(289,336)(290,335)(291,334)(292,342)(293,341)(294,340)(295,339)
(296,338)(297,337)(298,352)(299,354)(300,353)(301,358)(302,360)(303,359)
(304,355)(305,357)(306,356)(307,372)(308,371)(309,370)(310,378)(311,377)
(312,376)(313,375)(314,374)(315,373)(316,363)(317,362)(318,361)(319,369)
(320,368)(321,367)(322,366)(323,365)(324,364);
s1 := Sym(432)!( 1,229)( 2,231)( 3,230)( 4,226)( 5,228)( 6,227)( 7,232)
( 8,234)( 9,233)( 10,220)( 11,222)( 12,221)( 13,217)( 14,219)( 15,218)
( 16,223)( 17,225)( 18,224)( 19,240)( 20,239)( 21,238)( 22,237)( 23,236)
( 24,235)( 25,243)( 26,242)( 27,241)( 28,256)( 29,258)( 30,257)( 31,253)
( 32,255)( 33,254)( 34,259)( 35,261)( 36,260)( 37,247)( 38,249)( 39,248)
( 40,244)( 41,246)( 42,245)( 43,250)( 44,252)( 45,251)( 46,267)( 47,266)
( 48,265)( 49,264)( 50,263)( 51,262)( 52,270)( 53,269)( 54,268)( 55,310)
( 56,312)( 57,311)( 58,307)( 59,309)( 60,308)( 61,313)( 62,315)( 63,314)
( 64,301)( 65,303)( 66,302)( 67,298)( 68,300)( 69,299)( 70,304)( 71,306)
( 72,305)( 73,321)( 74,320)( 75,319)( 76,318)( 77,317)( 78,316)( 79,324)
( 80,323)( 81,322)( 82,283)( 83,285)( 84,284)( 85,280)( 86,282)( 87,281)
( 88,286)( 89,288)( 90,287)( 91,274)( 92,276)( 93,275)( 94,271)( 95,273)
( 96,272)( 97,277)( 98,279)( 99,278)(100,294)(101,293)(102,292)(103,291)
(104,290)(105,289)(106,297)(107,296)(108,295)(109,364)(110,366)(111,365)
(112,361)(113,363)(114,362)(115,367)(116,369)(117,368)(118,355)(119,357)
(120,356)(121,352)(122,354)(123,353)(124,358)(125,360)(126,359)(127,375)
(128,374)(129,373)(130,372)(131,371)(132,370)(133,378)(134,377)(135,376)
(136,337)(137,339)(138,338)(139,334)(140,336)(141,335)(142,340)(143,342)
(144,341)(145,328)(146,330)(147,329)(148,325)(149,327)(150,326)(151,331)
(152,333)(153,332)(154,348)(155,347)(156,346)(157,345)(158,344)(159,343)
(160,351)(161,350)(162,349)(163,391)(164,393)(165,392)(166,388)(167,390)
(168,389)(169,394)(170,396)(171,395)(172,382)(173,384)(174,383)(175,379)
(176,381)(177,380)(178,385)(179,387)(180,386)(181,402)(182,401)(183,400)
(184,399)(185,398)(186,397)(187,405)(188,404)(189,403)(190,418)(191,420)
(192,419)(193,415)(194,417)(195,416)(196,421)(197,423)(198,422)(199,409)
(200,411)(201,410)(202,406)(203,408)(204,407)(205,412)(206,414)(207,413)
(208,429)(209,428)(210,427)(211,426)(212,425)(213,424)(214,432)(215,431)
(216,430);
s2 := 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);
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*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*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*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope