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