Polytope of Type {9,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,24}*1728
if this polytope has a name.
Group : SmallGroup(1728,12315)
Rank : 3
Schlafli Type : {9,24}
Number of vertices, edges, etc : 36, 432, 96
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 24
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 : {9,8}*576, {3,24}*576
   4-fold quotients : {9,12}*432
   9-fold quotients : {3,8}*192
   12-fold quotients : {9,4}*144, {3,12}*144
   16-fold quotients : {9,6}*108
   24-fold quotients : {9,4}*72
   36-fold quotients : {3,4}*48
   48-fold quotients : {9,2}*36, {3,6}*36
   72-fold quotients : {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, 15)( 10, 16)( 11, 14)( 12, 13)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 47)( 26, 48)
( 27, 46)( 28, 45)( 29, 44)( 30, 43)( 31, 41)( 32, 42)( 49, 97)( 50, 98)
( 51,100)( 52, 99)( 53,101)( 54,102)( 55,104)( 56,103)( 57,111)( 58,112)
( 59,110)( 60,109)( 61,108)( 62,107)( 63,105)( 64,106)( 65,129)( 66,130)
( 67,132)( 68,131)( 69,133)( 70,134)( 71,136)( 72,135)( 73,143)( 74,144)
( 75,142)( 76,141)( 77,140)( 78,139)( 79,137)( 80,138)( 81,113)( 82,114)
( 83,116)( 84,115)( 85,117)( 86,118)( 87,120)( 88,119)( 89,127)( 90,128)
( 91,126)( 92,125)( 93,124)( 94,123)( 95,121)( 96,122)(145,321)(146,322)
(147,324)(148,323)(149,325)(150,326)(151,328)(152,327)(153,335)(154,336)
(155,334)(156,333)(157,332)(158,331)(159,329)(160,330)(161,305)(162,306)
(163,308)(164,307)(165,309)(166,310)(167,312)(168,311)(169,319)(170,320)
(171,318)(172,317)(173,316)(174,315)(175,313)(176,314)(177,289)(178,290)
(179,292)(180,291)(181,293)(182,294)(183,296)(184,295)(185,303)(186,304)
(187,302)(188,301)(189,300)(190,299)(191,297)(192,298)(193,417)(194,418)
(195,420)(196,419)(197,421)(198,422)(199,424)(200,423)(201,431)(202,432)
(203,430)(204,429)(205,428)(206,427)(207,425)(208,426)(209,401)(210,402)
(211,404)(212,403)(213,405)(214,406)(215,408)(216,407)(217,415)(218,416)
(219,414)(220,413)(221,412)(222,411)(223,409)(224,410)(225,385)(226,386)
(227,388)(228,387)(229,389)(230,390)(231,392)(232,391)(233,399)(234,400)
(235,398)(236,397)(237,396)(238,395)(239,393)(240,394)(241,369)(242,370)
(243,372)(244,371)(245,373)(246,374)(247,376)(248,375)(249,383)(250,384)
(251,382)(252,381)(253,380)(254,379)(255,377)(256,378)(257,353)(258,354)
(259,356)(260,355)(261,357)(262,358)(263,360)(264,359)(265,367)(266,368)
(267,366)(268,365)(269,364)(270,363)(271,361)(272,362)(273,337)(274,338)
(275,340)(276,339)(277,341)(278,342)(279,344)(280,343)(281,351)(282,352)
(283,350)(284,349)(285,348)(286,347)(287,345)(288,346);;
s1 := (  1,193)(  2,196)(  3,195)(  4,194)(  5,205)(  6,208)(  7,207)(  8,206)
(  9,203)( 10,202)( 11,201)( 12,204)( 13,197)( 14,200)( 15,199)( 16,198)
( 17,225)( 18,228)( 19,227)( 20,226)( 21,237)( 22,240)( 23,239)( 24,238)
( 25,235)( 26,234)( 27,233)( 28,236)( 29,229)( 30,232)( 31,231)( 32,230)
( 33,209)( 34,212)( 35,211)( 36,210)( 37,221)( 38,224)( 39,223)( 40,222)
( 41,219)( 42,218)( 43,217)( 44,220)( 45,213)( 46,216)( 47,215)( 48,214)
( 49,145)( 50,148)( 51,147)( 52,146)( 53,157)( 54,160)( 55,159)( 56,158)
( 57,155)( 58,154)( 59,153)( 60,156)( 61,149)( 62,152)( 63,151)( 64,150)
( 65,177)( 66,180)( 67,179)( 68,178)( 69,189)( 70,192)( 71,191)( 72,190)
( 73,187)( 74,186)( 75,185)( 76,188)( 77,181)( 78,184)( 79,183)( 80,182)
( 81,161)( 82,164)( 83,163)( 84,162)( 85,173)( 86,176)( 87,175)( 88,174)
( 89,171)( 90,170)( 91,169)( 92,172)( 93,165)( 94,168)( 95,167)( 96,166)
( 97,241)( 98,244)( 99,243)(100,242)(101,253)(102,256)(103,255)(104,254)
(105,251)(106,250)(107,249)(108,252)(109,245)(110,248)(111,247)(112,246)
(113,273)(114,276)(115,275)(116,274)(117,285)(118,288)(119,287)(120,286)
(121,283)(122,282)(123,281)(124,284)(125,277)(126,280)(127,279)(128,278)
(129,257)(130,260)(131,259)(132,258)(133,269)(134,272)(135,271)(136,270)
(137,267)(138,266)(139,265)(140,268)(141,261)(142,264)(143,263)(144,262)
(289,369)(290,372)(291,371)(292,370)(293,381)(294,384)(295,383)(296,382)
(297,379)(298,378)(299,377)(300,380)(301,373)(302,376)(303,375)(304,374)
(305,353)(306,356)(307,355)(308,354)(309,365)(310,368)(311,367)(312,366)
(313,363)(314,362)(315,361)(316,364)(317,357)(318,360)(319,359)(320,358)
(321,337)(322,340)(323,339)(324,338)(325,349)(326,352)(327,351)(328,350)
(329,347)(330,346)(331,345)(332,348)(333,341)(334,344)(335,343)(336,342)
(385,417)(386,420)(387,419)(388,418)(389,429)(390,432)(391,431)(392,430)
(393,427)(394,426)(395,425)(396,428)(397,421)(398,424)(399,423)(400,422)
(402,404)(405,413)(406,416)(407,415)(408,414)(409,411);;
s2 := (  1,  5)(  2,  6)(  3,  7)(  4,  8)(  9, 15)( 10, 16)( 11, 13)( 12, 14)
( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 31)( 26, 32)( 27, 29)( 28, 30)
( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 47)( 42, 48)( 43, 45)( 44, 46)
( 49,101)( 50,102)( 51,103)( 52,104)( 53, 97)( 54, 98)( 55, 99)( 56,100)
( 57,111)( 58,112)( 59,109)( 60,110)( 61,107)( 62,108)( 63,105)( 64,106)
( 65,117)( 66,118)( 67,119)( 68,120)( 69,113)( 70,114)( 71,115)( 72,116)
( 73,127)( 74,128)( 75,125)( 76,126)( 77,123)( 78,124)( 79,121)( 80,122)
( 81,133)( 82,134)( 83,135)( 84,136)( 85,129)( 86,130)( 87,131)( 88,132)
( 89,143)( 90,144)( 91,141)( 92,142)( 93,139)( 94,140)( 95,137)( 96,138)
(145,149)(146,150)(147,151)(148,152)(153,159)(154,160)(155,157)(156,158)
(161,165)(162,166)(163,167)(164,168)(169,175)(170,176)(171,173)(172,174)
(177,181)(178,182)(179,183)(180,184)(185,191)(186,192)(187,189)(188,190)
(193,245)(194,246)(195,247)(196,248)(197,241)(198,242)(199,243)(200,244)
(201,255)(202,256)(203,253)(204,254)(205,251)(206,252)(207,249)(208,250)
(209,261)(210,262)(211,263)(212,264)(213,257)(214,258)(215,259)(216,260)
(217,271)(218,272)(219,269)(220,270)(221,267)(222,268)(223,265)(224,266)
(225,277)(226,278)(227,279)(228,280)(229,273)(230,274)(231,275)(232,276)
(233,287)(234,288)(235,285)(236,286)(237,283)(238,284)(239,281)(240,282)
(289,293)(290,294)(291,295)(292,296)(297,303)(298,304)(299,301)(300,302)
(305,309)(306,310)(307,311)(308,312)(313,319)(314,320)(315,317)(316,318)
(321,325)(322,326)(323,327)(324,328)(329,335)(330,336)(331,333)(332,334)
(337,389)(338,390)(339,391)(340,392)(341,385)(342,386)(343,387)(344,388)
(345,399)(346,400)(347,397)(348,398)(349,395)(350,396)(351,393)(352,394)
(353,405)(354,406)(355,407)(356,408)(357,401)(358,402)(359,403)(360,404)
(361,415)(362,416)(363,413)(364,414)(365,411)(366,412)(367,409)(368,410)
(369,421)(370,422)(371,423)(372,424)(373,417)(374,418)(375,419)(376,420)
(377,431)(378,432)(379,429)(380,430)(381,427)(382,428)(383,425)(384,426);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope