Polytope of Type {12,36}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,36}*1728f
if this polytope has a name.
Group : SmallGroup(1728,30228)
Rank : 3
Schlafli Type : {12,36}
Number of vertices, edges, etc : 24, 432, 72
Order of s0s1s2 : 36
Order of s0s1s2s1 : 12
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 : {12,18}*864b
3-fold quotients : {4,36}*576b, {12,12}*576g
4-fold quotients : {6,36}*432b, {12,9}*432
6-fold quotients : {4,36}*288b, {4,36}*288c, {4,18}*288, {12,6}*288b
8-fold quotients : {6,18}*216b
9-fold quotients : {4,12}*192b
12-fold quotients : {2,36}*144, {4,9}*144, {4,18}*144b, {4,18}*144c, {6,12}*144b, {12,3}*144
16-fold quotients : {6,9}*108
18-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
24-fold quotients : {4,9}*72, {2,18}*72, {6,6}*72b
36-fold quotients : {2,12}*48, {4,3}*48, {4,6}*48b, {4,6}*48c
48-fold quotients : {2,9}*36, {6,3}*36
72-fold quotients : {4,3}*24, {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 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 27)( 14, 28)
( 15, 25)( 16, 26)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 35)( 22, 36)
( 23, 33)( 24, 34)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 67)( 54, 68)( 55, 65)( 56, 66)
( 57, 71)( 58, 72)( 59, 69)( 60, 70)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,103)( 90,104)
( 91,101)( 92,102)( 93,107)( 94,108)( 95,105)( 96,106)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,135)(122,136)(123,133)(124,134)
(125,139)(126,140)(127,137)(128,138)(129,143)(130,144)(131,141)(132,142)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,171)(158,172)
(159,169)(160,170)(161,175)(162,176)(163,173)(164,174)(165,179)(166,180)
(167,177)(168,178)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,207)(194,208)(195,205)(196,206)(197,211)(198,212)(199,209)(200,210)
(201,215)(202,216)(203,213)(204,214)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,243)(230,244)(231,241)(232,242)(233,247)(234,248)
(235,245)(236,246)(237,251)(238,252)(239,249)(240,250)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,279)(266,280)(267,277)(268,278)
(269,283)(270,284)(271,281)(272,282)(273,287)(274,288)(275,285)(276,286)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,315)(302,316)
(303,313)(304,314)(305,319)(306,320)(307,317)(308,318)(309,323)(310,324)
(311,321)(312,322)(325,327)(326,328)(329,331)(330,332)(333,335)(334,336)
(337,351)(338,352)(339,349)(340,350)(341,355)(342,356)(343,353)(344,354)
(345,359)(346,360)(347,357)(348,358)(361,363)(362,364)(365,367)(366,368)
(369,371)(370,372)(373,387)(374,388)(375,385)(376,386)(377,391)(378,392)
(379,389)(380,390)(381,395)(382,396)(383,393)(384,394)(397,399)(398,400)
(401,403)(402,404)(405,407)(406,408)(409,423)(410,424)(411,421)(412,422)
(413,427)(414,428)(415,425)(416,426)(417,431)(418,432)(419,429)(420,430);;
s1 := ( 1, 13)( 2, 14)( 3, 16)( 4, 15)( 5, 21)( 6, 22)( 7, 24)( 8, 23)
( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 27, 28)( 29, 33)( 30, 34)( 31, 36)
( 32, 35)( 37, 93)( 38, 94)( 39, 96)( 40, 95)( 41, 89)( 42, 90)( 43, 92)
( 44, 91)( 45, 85)( 46, 86)( 47, 88)( 48, 87)( 49, 81)( 50, 82)( 51, 84)
( 52, 83)( 53, 77)( 54, 78)( 55, 80)( 56, 79)( 57, 73)( 58, 74)( 59, 76)
( 60, 75)( 61,105)( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)( 67,104)
( 68,103)( 69, 97)( 70, 98)( 71,100)( 72, 99)(109,121)(110,122)(111,124)
(112,123)(113,129)(114,130)(115,132)(116,131)(117,125)(118,126)(119,128)
(120,127)(135,136)(137,141)(138,142)(139,144)(140,143)(145,201)(146,202)
(147,204)(148,203)(149,197)(150,198)(151,200)(152,199)(153,193)(154,194)
(155,196)(156,195)(157,189)(158,190)(159,192)(160,191)(161,185)(162,186)
(163,188)(164,187)(165,181)(166,182)(167,184)(168,183)(169,213)(170,214)
(171,216)(172,215)(173,209)(174,210)(175,212)(176,211)(177,205)(178,206)
(179,208)(180,207)(217,337)(218,338)(219,340)(220,339)(221,345)(222,346)
(223,348)(224,347)(225,341)(226,342)(227,344)(228,343)(229,325)(230,326)
(231,328)(232,327)(233,333)(234,334)(235,336)(236,335)(237,329)(238,330)
(239,332)(240,331)(241,349)(242,350)(243,352)(244,351)(245,357)(246,358)
(247,360)(248,359)(249,353)(250,354)(251,356)(252,355)(253,417)(254,418)
(255,420)(256,419)(257,413)(258,414)(259,416)(260,415)(261,409)(262,410)
(263,412)(264,411)(265,405)(266,406)(267,408)(268,407)(269,401)(270,402)
(271,404)(272,403)(273,397)(274,398)(275,400)(276,399)(277,429)(278,430)
(279,432)(280,431)(281,425)(282,426)(283,428)(284,427)(285,421)(286,422)
(287,424)(288,423)(289,381)(290,382)(291,384)(292,383)(293,377)(294,378)
(295,380)(296,379)(297,373)(298,374)(299,376)(300,375)(301,369)(302,370)
(303,372)(304,371)(305,365)(306,366)(307,368)(308,367)(309,361)(310,362)
(311,364)(312,363)(313,393)(314,394)(315,396)(316,395)(317,389)(318,390)
(319,392)(320,391)(321,385)(322,386)(323,388)(324,387);;
s2 := ( 1,253)( 2,256)( 3,255)( 4,254)( 5,261)( 6,264)( 7,263)( 8,262)
( 9,257)( 10,260)( 11,259)( 12,258)( 13,277)( 14,280)( 15,279)( 16,278)
( 17,285)( 18,288)( 19,287)( 20,286)( 21,281)( 22,284)( 23,283)( 24,282)
( 25,265)( 26,268)( 27,267)( 28,266)( 29,273)( 30,276)( 31,275)( 32,274)
( 33,269)( 34,272)( 35,271)( 36,270)( 37,217)( 38,220)( 39,219)( 40,218)
( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)( 48,222)
( 49,241)( 50,244)( 51,243)( 52,242)( 53,249)( 54,252)( 55,251)( 56,250)
( 57,245)( 58,248)( 59,247)( 60,246)( 61,229)( 62,232)( 63,231)( 64,230)
( 65,237)( 66,240)( 67,239)( 68,238)( 69,233)( 70,236)( 71,235)( 72,234)
( 73,297)( 74,300)( 75,299)( 76,298)( 77,293)( 78,296)( 79,295)( 80,294)
( 81,289)( 82,292)( 83,291)( 84,290)( 85,321)( 86,324)( 87,323)( 88,322)
( 89,317)( 90,320)( 91,319)( 92,318)( 93,313)( 94,316)( 95,315)( 96,314)
( 97,309)( 98,312)( 99,311)(100,310)(101,305)(102,308)(103,307)(104,306)
(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)(112,362)
(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)(120,366)
(121,385)(122,388)(123,387)(124,386)(125,393)(126,396)(127,395)(128,394)
(129,389)(130,392)(131,391)(132,390)(133,373)(134,376)(135,375)(136,374)
(137,381)(138,384)(139,383)(140,382)(141,377)(142,380)(143,379)(144,378)
(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)(152,334)
(153,329)(154,332)(155,331)(156,330)(157,349)(158,352)(159,351)(160,350)
(161,357)(162,360)(163,359)(164,358)(165,353)(166,356)(167,355)(168,354)
(169,337)(170,340)(171,339)(172,338)(173,345)(174,348)(175,347)(176,346)
(177,341)(178,344)(179,343)(180,342)(181,405)(182,408)(183,407)(184,406)
(185,401)(186,404)(187,403)(188,402)(189,397)(190,400)(191,399)(192,398)
(193,429)(194,432)(195,431)(196,430)(197,425)(198,428)(199,427)(200,426)
(201,421)(202,424)(203,423)(204,422)(205,417)(206,420)(207,419)(208,418)
(209,413)(210,416)(211,415)(212,414)(213,409)(214,412)(215,411)(216,410);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*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)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 27)
( 14, 28)( 15, 25)( 16, 26)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 35)
( 22, 36)( 23, 33)( 24, 34)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 67)( 54, 68)( 55, 65)
( 56, 66)( 57, 71)( 58, 72)( 59, 69)( 60, 70)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,103)
( 90,104)( 91,101)( 92,102)( 93,107)( 94,108)( 95,105)( 96,106)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,135)(122,136)(123,133)
(124,134)(125,139)(126,140)(127,137)(128,138)(129,143)(130,144)(131,141)
(132,142)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,171)
(158,172)(159,169)(160,170)(161,175)(162,176)(163,173)(164,174)(165,179)
(166,180)(167,177)(168,178)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,207)(194,208)(195,205)(196,206)(197,211)(198,212)(199,209)
(200,210)(201,215)(202,216)(203,213)(204,214)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,243)(230,244)(231,241)(232,242)(233,247)
(234,248)(235,245)(236,246)(237,251)(238,252)(239,249)(240,250)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,279)(266,280)(267,277)
(268,278)(269,283)(270,284)(271,281)(272,282)(273,287)(274,288)(275,285)
(276,286)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,315)
(302,316)(303,313)(304,314)(305,319)(306,320)(307,317)(308,318)(309,323)
(310,324)(311,321)(312,322)(325,327)(326,328)(329,331)(330,332)(333,335)
(334,336)(337,351)(338,352)(339,349)(340,350)(341,355)(342,356)(343,353)
(344,354)(345,359)(346,360)(347,357)(348,358)(361,363)(362,364)(365,367)
(366,368)(369,371)(370,372)(373,387)(374,388)(375,385)(376,386)(377,391)
(378,392)(379,389)(380,390)(381,395)(382,396)(383,393)(384,394)(397,399)
(398,400)(401,403)(402,404)(405,407)(406,408)(409,423)(410,424)(411,421)
(412,422)(413,427)(414,428)(415,425)(416,426)(417,431)(418,432)(419,429)
(420,430);
s1 := Sym(432)!( 1, 13)( 2, 14)( 3, 16)( 4, 15)( 5, 21)( 6, 22)( 7, 24)
( 8, 23)( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 27, 28)( 29, 33)( 30, 34)
( 31, 36)( 32, 35)( 37, 93)( 38, 94)( 39, 96)( 40, 95)( 41, 89)( 42, 90)
( 43, 92)( 44, 91)( 45, 85)( 46, 86)( 47, 88)( 48, 87)( 49, 81)( 50, 82)
( 51, 84)( 52, 83)( 53, 77)( 54, 78)( 55, 80)( 56, 79)( 57, 73)( 58, 74)
( 59, 76)( 60, 75)( 61,105)( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)
( 67,104)( 68,103)( 69, 97)( 70, 98)( 71,100)( 72, 99)(109,121)(110,122)
(111,124)(112,123)(113,129)(114,130)(115,132)(116,131)(117,125)(118,126)
(119,128)(120,127)(135,136)(137,141)(138,142)(139,144)(140,143)(145,201)
(146,202)(147,204)(148,203)(149,197)(150,198)(151,200)(152,199)(153,193)
(154,194)(155,196)(156,195)(157,189)(158,190)(159,192)(160,191)(161,185)
(162,186)(163,188)(164,187)(165,181)(166,182)(167,184)(168,183)(169,213)
(170,214)(171,216)(172,215)(173,209)(174,210)(175,212)(176,211)(177,205)
(178,206)(179,208)(180,207)(217,337)(218,338)(219,340)(220,339)(221,345)
(222,346)(223,348)(224,347)(225,341)(226,342)(227,344)(228,343)(229,325)
(230,326)(231,328)(232,327)(233,333)(234,334)(235,336)(236,335)(237,329)
(238,330)(239,332)(240,331)(241,349)(242,350)(243,352)(244,351)(245,357)
(246,358)(247,360)(248,359)(249,353)(250,354)(251,356)(252,355)(253,417)
(254,418)(255,420)(256,419)(257,413)(258,414)(259,416)(260,415)(261,409)
(262,410)(263,412)(264,411)(265,405)(266,406)(267,408)(268,407)(269,401)
(270,402)(271,404)(272,403)(273,397)(274,398)(275,400)(276,399)(277,429)
(278,430)(279,432)(280,431)(281,425)(282,426)(283,428)(284,427)(285,421)
(286,422)(287,424)(288,423)(289,381)(290,382)(291,384)(292,383)(293,377)
(294,378)(295,380)(296,379)(297,373)(298,374)(299,376)(300,375)(301,369)
(302,370)(303,372)(304,371)(305,365)(306,366)(307,368)(308,367)(309,361)
(310,362)(311,364)(312,363)(313,393)(314,394)(315,396)(316,395)(317,389)
(318,390)(319,392)(320,391)(321,385)(322,386)(323,388)(324,387);
s2 := Sym(432)!( 1,253)( 2,256)( 3,255)( 4,254)( 5,261)( 6,264)( 7,263)
( 8,262)( 9,257)( 10,260)( 11,259)( 12,258)( 13,277)( 14,280)( 15,279)
( 16,278)( 17,285)( 18,288)( 19,287)( 20,286)( 21,281)( 22,284)( 23,283)
( 24,282)( 25,265)( 26,268)( 27,267)( 28,266)( 29,273)( 30,276)( 31,275)
( 32,274)( 33,269)( 34,272)( 35,271)( 36,270)( 37,217)( 38,220)( 39,219)
( 40,218)( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)
( 48,222)( 49,241)( 50,244)( 51,243)( 52,242)( 53,249)( 54,252)( 55,251)
( 56,250)( 57,245)( 58,248)( 59,247)( 60,246)( 61,229)( 62,232)( 63,231)
( 64,230)( 65,237)( 66,240)( 67,239)( 68,238)( 69,233)( 70,236)( 71,235)
( 72,234)( 73,297)( 74,300)( 75,299)( 76,298)( 77,293)( 78,296)( 79,295)
( 80,294)( 81,289)( 82,292)( 83,291)( 84,290)( 85,321)( 86,324)( 87,323)
( 88,322)( 89,317)( 90,320)( 91,319)( 92,318)( 93,313)( 94,316)( 95,315)
( 96,314)( 97,309)( 98,312)( 99,311)(100,310)(101,305)(102,308)(103,307)
(104,306)(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)
(112,362)(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)
(120,366)(121,385)(122,388)(123,387)(124,386)(125,393)(126,396)(127,395)
(128,394)(129,389)(130,392)(131,391)(132,390)(133,373)(134,376)(135,375)
(136,374)(137,381)(138,384)(139,383)(140,382)(141,377)(142,380)(143,379)
(144,378)(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)
(152,334)(153,329)(154,332)(155,331)(156,330)(157,349)(158,352)(159,351)
(160,350)(161,357)(162,360)(163,359)(164,358)(165,353)(166,356)(167,355)
(168,354)(169,337)(170,340)(171,339)(172,338)(173,345)(174,348)(175,347)
(176,346)(177,341)(178,344)(179,343)(180,342)(181,405)(182,408)(183,407)
(184,406)(185,401)(186,404)(187,403)(188,402)(189,397)(190,400)(191,399)
(192,398)(193,429)(194,432)(195,431)(196,430)(197,425)(198,428)(199,427)
(200,426)(201,421)(202,424)(203,423)(204,422)(205,417)(206,420)(207,419)
(208,418)(209,413)(210,416)(211,415)(212,414)(213,409)(214,412)(215,411)
(216,410);
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*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