Polytope of Type {9,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,12}*1728
if this polytope has a name.
Group : SmallGroup(1728,46098)
Rank : 3
Schlafli Type : {9,12}
Number of vertices, edges, etc : 72, 432, 96
Order of s0s1s2 : 36
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {3,12}*576
4-fold quotients : {9,6}*432, {9,12}*432
12-fold quotients : {9,4}*144, {3,6}*144, {3,12}*144
16-fold quotients : {9,6}*108
24-fold quotients : {9,4}*72
36-fold quotients : {3,4}*48, {3,6}*48
48-fold quotients : {9,2}*36, {3,6}*36
72-fold quotients : {3,3}*24, {3,4}*24
144-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 49, 97)( 50, 98)
( 51,100)( 52, 99)( 53,101)( 54,102)( 55,104)( 56,103)( 57,109)( 58,110)
( 59,112)( 60,111)( 61,105)( 62,106)( 63,108)( 64,107)( 65,129)( 66,130)
( 67,132)( 68,131)( 69,133)( 70,134)( 71,136)( 72,135)( 73,141)( 74,142)
( 75,144)( 76,143)( 77,137)( 78,138)( 79,140)( 80,139)( 81,113)( 82,114)
( 83,116)( 84,115)( 85,117)( 86,118)( 87,120)( 88,119)( 89,125)( 90,126)
( 91,128)( 92,127)( 93,121)( 94,122)( 95,124)( 96,123)(145,321)(146,322)
(147,324)(148,323)(149,325)(150,326)(151,328)(152,327)(153,333)(154,334)
(155,336)(156,335)(157,329)(158,330)(159,332)(160,331)(161,305)(162,306)
(163,308)(164,307)(165,309)(166,310)(167,312)(168,311)(169,317)(170,318)
(171,320)(172,319)(173,313)(174,314)(175,316)(176,315)(177,289)(178,290)
(179,292)(180,291)(181,293)(182,294)(183,296)(184,295)(185,301)(186,302)
(187,304)(188,303)(189,297)(190,298)(191,300)(192,299)(193,417)(194,418)
(195,420)(196,419)(197,421)(198,422)(199,424)(200,423)(201,429)(202,430)
(203,432)(204,431)(205,425)(206,426)(207,428)(208,427)(209,401)(210,402)
(211,404)(212,403)(213,405)(214,406)(215,408)(216,407)(217,413)(218,414)
(219,416)(220,415)(221,409)(222,410)(223,412)(224,411)(225,385)(226,386)
(227,388)(228,387)(229,389)(230,390)(231,392)(232,391)(233,397)(234,398)
(235,400)(236,399)(237,393)(238,394)(239,396)(240,395)(241,369)(242,370)
(243,372)(244,371)(245,373)(246,374)(247,376)(248,375)(249,381)(250,382)
(251,384)(252,383)(253,377)(254,378)(255,380)(256,379)(257,353)(258,354)
(259,356)(260,355)(261,357)(262,358)(263,360)(264,359)(265,365)(266,366)
(267,368)(268,367)(269,361)(270,362)(271,364)(272,363)(273,337)(274,338)
(275,340)(276,339)(277,341)(278,342)(279,344)(280,343)(281,349)(282,350)
(283,352)(284,351)(285,345)(286,346)(287,348)(288,347);;
s1 := ( 1,193)( 2,196)( 3,195)( 4,194)( 5,205)( 6,208)( 7,207)( 8,206)
( 9,201)( 10,204)( 11,203)( 12,202)( 13,197)( 14,200)( 15,199)( 16,198)
( 17,225)( 18,228)( 19,227)( 20,226)( 21,237)( 22,240)( 23,239)( 24,238)
( 25,233)( 26,236)( 27,235)( 28,234)( 29,229)( 30,232)( 31,231)( 32,230)
( 33,209)( 34,212)( 35,211)( 36,210)( 37,221)( 38,224)( 39,223)( 40,222)
( 41,217)( 42,220)( 43,219)( 44,218)( 45,213)( 46,216)( 47,215)( 48,214)
( 49,145)( 50,148)( 51,147)( 52,146)( 53,157)( 54,160)( 55,159)( 56,158)
( 57,153)( 58,156)( 59,155)( 60,154)( 61,149)( 62,152)( 63,151)( 64,150)
( 65,177)( 66,180)( 67,179)( 68,178)( 69,189)( 70,192)( 71,191)( 72,190)
( 73,185)( 74,188)( 75,187)( 76,186)( 77,181)( 78,184)( 79,183)( 80,182)
( 81,161)( 82,164)( 83,163)( 84,162)( 85,173)( 86,176)( 87,175)( 88,174)
( 89,169)( 90,172)( 91,171)( 92,170)( 93,165)( 94,168)( 95,167)( 96,166)
( 97,241)( 98,244)( 99,243)(100,242)(101,253)(102,256)(103,255)(104,254)
(105,249)(106,252)(107,251)(108,250)(109,245)(110,248)(111,247)(112,246)
(113,273)(114,276)(115,275)(116,274)(117,285)(118,288)(119,287)(120,286)
(121,281)(122,284)(123,283)(124,282)(125,277)(126,280)(127,279)(128,278)
(129,257)(130,260)(131,259)(132,258)(133,269)(134,272)(135,271)(136,270)
(137,265)(138,268)(139,267)(140,266)(141,261)(142,264)(143,263)(144,262)
(289,369)(290,372)(291,371)(292,370)(293,381)(294,384)(295,383)(296,382)
(297,377)(298,380)(299,379)(300,378)(301,373)(302,376)(303,375)(304,374)
(305,353)(306,356)(307,355)(308,354)(309,365)(310,368)(311,367)(312,366)
(313,361)(314,364)(315,363)(316,362)(317,357)(318,360)(319,359)(320,358)
(321,337)(322,340)(323,339)(324,338)(325,349)(326,352)(327,351)(328,350)
(329,345)(330,348)(331,347)(332,346)(333,341)(334,344)(335,343)(336,342)
(385,417)(386,420)(387,419)(388,418)(389,429)(390,432)(391,431)(392,430)
(393,425)(394,428)(395,427)(396,426)(397,421)(398,424)(399,423)(400,422)
(402,404)(405,413)(406,416)(407,415)(408,414)(410,412);;
s2 := ( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 22)( 18, 21)( 19, 24)( 20, 23)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 38)( 34, 37)( 35, 40)( 36, 39)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49,102)( 50,101)( 51,104)( 52,103)( 53, 98)( 54, 97)( 55,100)( 56, 99)
( 57,106)( 58,105)( 59,108)( 60,107)( 61,110)( 62,109)( 63,112)( 64,111)
( 65,118)( 66,117)( 67,120)( 68,119)( 69,114)( 70,113)( 71,116)( 72,115)
( 73,122)( 74,121)( 75,124)( 76,123)( 77,126)( 78,125)( 79,128)( 80,127)
( 81,134)( 82,133)( 83,136)( 84,135)( 85,130)( 86,129)( 87,132)( 88,131)
( 89,138)( 90,137)( 91,140)( 92,139)( 93,142)( 94,141)( 95,144)( 96,143)
(145,150)(146,149)(147,152)(148,151)(153,154)(155,156)(157,158)(159,160)
(161,166)(162,165)(163,168)(164,167)(169,170)(171,172)(173,174)(175,176)
(177,182)(178,181)(179,184)(180,183)(185,186)(187,188)(189,190)(191,192)
(193,246)(194,245)(195,248)(196,247)(197,242)(198,241)(199,244)(200,243)
(201,250)(202,249)(203,252)(204,251)(205,254)(206,253)(207,256)(208,255)
(209,262)(210,261)(211,264)(212,263)(213,258)(214,257)(215,260)(216,259)
(217,266)(218,265)(219,268)(220,267)(221,270)(222,269)(223,272)(224,271)
(225,278)(226,277)(227,280)(228,279)(229,274)(230,273)(231,276)(232,275)
(233,282)(234,281)(235,284)(236,283)(237,286)(238,285)(239,288)(240,287)
(289,294)(290,293)(291,296)(292,295)(297,298)(299,300)(301,302)(303,304)
(305,310)(306,309)(307,312)(308,311)(313,314)(315,316)(317,318)(319,320)
(321,326)(322,325)(323,328)(324,327)(329,330)(331,332)(333,334)(335,336)
(337,390)(338,389)(339,392)(340,391)(341,386)(342,385)(343,388)(344,387)
(345,394)(346,393)(347,396)(348,395)(349,398)(350,397)(351,400)(352,399)
(353,406)(354,405)(355,408)(356,407)(357,402)(358,401)(359,404)(360,403)
(361,410)(362,409)(363,412)(364,411)(365,414)(366,413)(367,416)(368,415)
(369,422)(370,421)(371,424)(372,423)(373,418)(374,417)(375,420)(376,419)
(377,426)(378,425)(379,428)(380,427)(381,430)(382,429)(383,432)(384,431);;
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*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,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 49, 97)
( 50, 98)( 51,100)( 52, 99)( 53,101)( 54,102)( 55,104)( 56,103)( 57,109)
( 58,110)( 59,112)( 60,111)( 61,105)( 62,106)( 63,108)( 64,107)( 65,129)
( 66,130)( 67,132)( 68,131)( 69,133)( 70,134)( 71,136)( 72,135)( 73,141)
( 74,142)( 75,144)( 76,143)( 77,137)( 78,138)( 79,140)( 80,139)( 81,113)
( 82,114)( 83,116)( 84,115)( 85,117)( 86,118)( 87,120)( 88,119)( 89,125)
( 90,126)( 91,128)( 92,127)( 93,121)( 94,122)( 95,124)( 96,123)(145,321)
(146,322)(147,324)(148,323)(149,325)(150,326)(151,328)(152,327)(153,333)
(154,334)(155,336)(156,335)(157,329)(158,330)(159,332)(160,331)(161,305)
(162,306)(163,308)(164,307)(165,309)(166,310)(167,312)(168,311)(169,317)
(170,318)(171,320)(172,319)(173,313)(174,314)(175,316)(176,315)(177,289)
(178,290)(179,292)(180,291)(181,293)(182,294)(183,296)(184,295)(185,301)
(186,302)(187,304)(188,303)(189,297)(190,298)(191,300)(192,299)(193,417)
(194,418)(195,420)(196,419)(197,421)(198,422)(199,424)(200,423)(201,429)
(202,430)(203,432)(204,431)(205,425)(206,426)(207,428)(208,427)(209,401)
(210,402)(211,404)(212,403)(213,405)(214,406)(215,408)(216,407)(217,413)
(218,414)(219,416)(220,415)(221,409)(222,410)(223,412)(224,411)(225,385)
(226,386)(227,388)(228,387)(229,389)(230,390)(231,392)(232,391)(233,397)
(234,398)(235,400)(236,399)(237,393)(238,394)(239,396)(240,395)(241,369)
(242,370)(243,372)(244,371)(245,373)(246,374)(247,376)(248,375)(249,381)
(250,382)(251,384)(252,383)(253,377)(254,378)(255,380)(256,379)(257,353)
(258,354)(259,356)(260,355)(261,357)(262,358)(263,360)(264,359)(265,365)
(266,366)(267,368)(268,367)(269,361)(270,362)(271,364)(272,363)(273,337)
(274,338)(275,340)(276,339)(277,341)(278,342)(279,344)(280,343)(281,349)
(282,350)(283,352)(284,351)(285,345)(286,346)(287,348)(288,347);
s1 := Sym(432)!( 1,193)( 2,196)( 3,195)( 4,194)( 5,205)( 6,208)( 7,207)
( 8,206)( 9,201)( 10,204)( 11,203)( 12,202)( 13,197)( 14,200)( 15,199)
( 16,198)( 17,225)( 18,228)( 19,227)( 20,226)( 21,237)( 22,240)( 23,239)
( 24,238)( 25,233)( 26,236)( 27,235)( 28,234)( 29,229)( 30,232)( 31,231)
( 32,230)( 33,209)( 34,212)( 35,211)( 36,210)( 37,221)( 38,224)( 39,223)
( 40,222)( 41,217)( 42,220)( 43,219)( 44,218)( 45,213)( 46,216)( 47,215)
( 48,214)( 49,145)( 50,148)( 51,147)( 52,146)( 53,157)( 54,160)( 55,159)
( 56,158)( 57,153)( 58,156)( 59,155)( 60,154)( 61,149)( 62,152)( 63,151)
( 64,150)( 65,177)( 66,180)( 67,179)( 68,178)( 69,189)( 70,192)( 71,191)
( 72,190)( 73,185)( 74,188)( 75,187)( 76,186)( 77,181)( 78,184)( 79,183)
( 80,182)( 81,161)( 82,164)( 83,163)( 84,162)( 85,173)( 86,176)( 87,175)
( 88,174)( 89,169)( 90,172)( 91,171)( 92,170)( 93,165)( 94,168)( 95,167)
( 96,166)( 97,241)( 98,244)( 99,243)(100,242)(101,253)(102,256)(103,255)
(104,254)(105,249)(106,252)(107,251)(108,250)(109,245)(110,248)(111,247)
(112,246)(113,273)(114,276)(115,275)(116,274)(117,285)(118,288)(119,287)
(120,286)(121,281)(122,284)(123,283)(124,282)(125,277)(126,280)(127,279)
(128,278)(129,257)(130,260)(131,259)(132,258)(133,269)(134,272)(135,271)
(136,270)(137,265)(138,268)(139,267)(140,266)(141,261)(142,264)(143,263)
(144,262)(289,369)(290,372)(291,371)(292,370)(293,381)(294,384)(295,383)
(296,382)(297,377)(298,380)(299,379)(300,378)(301,373)(302,376)(303,375)
(304,374)(305,353)(306,356)(307,355)(308,354)(309,365)(310,368)(311,367)
(312,366)(313,361)(314,364)(315,363)(316,362)(317,357)(318,360)(319,359)
(320,358)(321,337)(322,340)(323,339)(324,338)(325,349)(326,352)(327,351)
(328,350)(329,345)(330,348)(331,347)(332,346)(333,341)(334,344)(335,343)
(336,342)(385,417)(386,420)(387,419)(388,418)(389,429)(390,432)(391,431)
(392,430)(393,425)(394,428)(395,427)(396,426)(397,421)(398,424)(399,423)
(400,422)(402,404)(405,413)(406,416)(407,415)(408,414)(410,412);
s2 := Sym(432)!( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 22)( 18, 21)( 19, 24)( 20, 23)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 38)( 34, 37)( 35, 40)( 36, 39)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49,102)( 50,101)( 51,104)( 52,103)( 53, 98)( 54, 97)( 55,100)
( 56, 99)( 57,106)( 58,105)( 59,108)( 60,107)( 61,110)( 62,109)( 63,112)
( 64,111)( 65,118)( 66,117)( 67,120)( 68,119)( 69,114)( 70,113)( 71,116)
( 72,115)( 73,122)( 74,121)( 75,124)( 76,123)( 77,126)( 78,125)( 79,128)
( 80,127)( 81,134)( 82,133)( 83,136)( 84,135)( 85,130)( 86,129)( 87,132)
( 88,131)( 89,138)( 90,137)( 91,140)( 92,139)( 93,142)( 94,141)( 95,144)
( 96,143)(145,150)(146,149)(147,152)(148,151)(153,154)(155,156)(157,158)
(159,160)(161,166)(162,165)(163,168)(164,167)(169,170)(171,172)(173,174)
(175,176)(177,182)(178,181)(179,184)(180,183)(185,186)(187,188)(189,190)
(191,192)(193,246)(194,245)(195,248)(196,247)(197,242)(198,241)(199,244)
(200,243)(201,250)(202,249)(203,252)(204,251)(205,254)(206,253)(207,256)
(208,255)(209,262)(210,261)(211,264)(212,263)(213,258)(214,257)(215,260)
(216,259)(217,266)(218,265)(219,268)(220,267)(221,270)(222,269)(223,272)
(224,271)(225,278)(226,277)(227,280)(228,279)(229,274)(230,273)(231,276)
(232,275)(233,282)(234,281)(235,284)(236,283)(237,286)(238,285)(239,288)
(240,287)(289,294)(290,293)(291,296)(292,295)(297,298)(299,300)(301,302)
(303,304)(305,310)(306,309)(307,312)(308,311)(313,314)(315,316)(317,318)
(319,320)(321,326)(322,325)(323,328)(324,327)(329,330)(331,332)(333,334)
(335,336)(337,390)(338,389)(339,392)(340,391)(341,386)(342,385)(343,388)
(344,387)(345,394)(346,393)(347,396)(348,395)(349,398)(350,397)(351,400)
(352,399)(353,406)(354,405)(355,408)(356,407)(357,402)(358,401)(359,404)
(360,403)(361,410)(362,409)(363,412)(364,411)(365,414)(366,413)(367,416)
(368,415)(369,422)(370,421)(371,424)(372,423)(373,418)(374,417)(375,420)
(376,419)(377,426)(378,425)(379,428)(380,427)(381,430)(382,429)(383,432)
(384,431);
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*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,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*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 >;
References : None.
to this polytope