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