Polytope of Type {18,48}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,48}*1728b
if this polytope has a name.
Group : SmallGroup(1728,5288)
Rank : 3
Schlafli Type : {18,48}
Number of vertices, edges, etc : 18, 432, 48
Order of s0s1s2 : 144
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
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 : {18,24}*864b
3-fold quotients : {18,16}*576, {6,48}*576c
4-fold quotients : {18,12}*432b
6-fold quotients : {18,8}*288, {6,24}*288c
8-fold quotients : {18,6}*216b
9-fold quotients : {6,16}*192
12-fold quotients : {18,4}*144a, {6,12}*144c
16-fold quotients : {9,6}*108
18-fold quotients : {6,8}*96
24-fold quotients : {18,2}*72, {6,6}*72c
27-fold quotients : {2,16}*64
36-fold quotients : {6,4}*48a
48-fold quotients : {9,2}*36, {3,6}*36
54-fold quotients : {2,8}*32
72-fold quotients : {6,2}*24
108-fold quotients : {2,4}*16
144-fold quotients : {3,2}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 10, 21)( 11, 20)( 12, 19)( 13, 27)
( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 48)( 38, 47)( 39, 46)( 40, 54)( 41, 53)( 42, 52)( 43, 51)
( 44, 50)( 45, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 75)( 65, 74)
( 66, 73)( 67, 81)( 68, 80)( 69, 79)( 70, 78)( 71, 77)( 72, 76)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 91,102)( 92,101)( 93,100)( 94,108)( 95,107)
( 96,106)( 97,105)( 98,104)( 99,103)(110,111)(112,115)(113,117)(114,116)
(118,129)(119,128)(120,127)(121,135)(122,134)(123,133)(124,132)(125,131)
(126,130)(137,138)(139,142)(140,144)(141,143)(145,156)(146,155)(147,154)
(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(164,165)(166,169)
(167,171)(168,170)(172,183)(173,182)(174,181)(175,189)(176,188)(177,187)
(178,186)(179,185)(180,184)(191,192)(193,196)(194,198)(195,197)(199,210)
(200,209)(201,208)(202,216)(203,215)(204,214)(205,213)(206,212)(207,211)
(218,219)(220,223)(221,225)(222,224)(226,237)(227,236)(228,235)(229,243)
(230,242)(231,241)(232,240)(233,239)(234,238)(245,246)(247,250)(248,252)
(249,251)(253,264)(254,263)(255,262)(256,270)(257,269)(258,268)(259,267)
(260,266)(261,265)(272,273)(274,277)(275,279)(276,278)(280,291)(281,290)
(282,289)(283,297)(284,296)(285,295)(286,294)(287,293)(288,292)(299,300)
(301,304)(302,306)(303,305)(307,318)(308,317)(309,316)(310,324)(311,323)
(312,322)(313,321)(314,320)(315,319)(326,327)(328,331)(329,333)(330,332)
(334,345)(335,344)(336,343)(337,351)(338,350)(339,349)(340,348)(341,347)
(342,346)(353,354)(355,358)(356,360)(357,359)(361,372)(362,371)(363,370)
(364,378)(365,377)(366,376)(367,375)(368,374)(369,373)(380,381)(382,385)
(383,387)(384,386)(388,399)(389,398)(390,397)(391,405)(392,404)(393,403)
(394,402)(395,401)(396,400)(407,408)(409,412)(410,414)(411,413)(415,426)
(416,425)(417,424)(418,432)(419,431)(420,430)(421,429)(422,428)(423,427);;
s1 := ( 1,229)( 2,231)( 3,230)( 4,226)( 5,228)( 6,227)( 7,232)( 8,234)
( 9,233)( 10,220)( 11,222)( 12,221)( 13,217)( 14,219)( 15,218)( 16,223)
( 17,225)( 18,224)( 19,240)( 20,239)( 21,238)( 22,237)( 23,236)( 24,235)
( 25,243)( 26,242)( 27,241)( 28,256)( 29,258)( 30,257)( 31,253)( 32,255)
( 33,254)( 34,259)( 35,261)( 36,260)( 37,247)( 38,249)( 39,248)( 40,244)
( 41,246)( 42,245)( 43,250)( 44,252)( 45,251)( 46,267)( 47,266)( 48,265)
( 49,264)( 50,263)( 51,262)( 52,270)( 53,269)( 54,268)( 55,310)( 56,312)
( 57,311)( 58,307)( 59,309)( 60,308)( 61,313)( 62,315)( 63,314)( 64,301)
( 65,303)( 66,302)( 67,298)( 68,300)( 69,299)( 70,304)( 71,306)( 72,305)
( 73,321)( 74,320)( 75,319)( 76,318)( 77,317)( 78,316)( 79,324)( 80,323)
( 81,322)( 82,283)( 83,285)( 84,284)( 85,280)( 86,282)( 87,281)( 88,286)
( 89,288)( 90,287)( 91,274)( 92,276)( 93,275)( 94,271)( 95,273)( 96,272)
( 97,277)( 98,279)( 99,278)(100,294)(101,293)(102,292)(103,291)(104,290)
(105,289)(106,297)(107,296)(108,295)(109,391)(110,393)(111,392)(112,388)
(113,390)(114,389)(115,394)(116,396)(117,395)(118,382)(119,384)(120,383)
(121,379)(122,381)(123,380)(124,385)(125,387)(126,386)(127,402)(128,401)
(129,400)(130,399)(131,398)(132,397)(133,405)(134,404)(135,403)(136,418)
(137,420)(138,419)(139,415)(140,417)(141,416)(142,421)(143,423)(144,422)
(145,409)(146,411)(147,410)(148,406)(149,408)(150,407)(151,412)(152,414)
(153,413)(154,429)(155,428)(156,427)(157,426)(158,425)(159,424)(160,432)
(161,431)(162,430)(163,337)(164,339)(165,338)(166,334)(167,336)(168,335)
(169,340)(170,342)(171,341)(172,328)(173,330)(174,329)(175,325)(176,327)
(177,326)(178,331)(179,333)(180,332)(181,348)(182,347)(183,346)(184,345)
(185,344)(186,343)(187,351)(188,350)(189,349)(190,364)(191,366)(192,365)
(193,361)(194,363)(195,362)(196,367)(197,369)(198,368)(199,355)(200,357)
(201,356)(202,352)(203,354)(204,353)(205,358)(206,360)(207,359)(208,375)
(209,374)(210,373)(211,372)(212,371)(213,370)(214,378)(215,377)(216,376);;
s2 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)
( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)( 68, 98)
( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)( 76,106)
( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(109,163)(110,164)(111,165)
(112,169)(113,170)(114,171)(115,166)(116,167)(117,168)(118,172)(119,173)
(120,174)(121,178)(122,179)(123,180)(124,175)(125,176)(126,177)(127,181)
(128,182)(129,183)(130,187)(131,188)(132,189)(133,184)(134,185)(135,186)
(136,190)(137,191)(138,192)(139,196)(140,197)(141,198)(142,193)(143,194)
(144,195)(145,199)(146,200)(147,201)(148,205)(149,206)(150,207)(151,202)
(152,203)(153,204)(154,208)(155,209)(156,210)(157,214)(158,215)(159,216)
(160,211)(161,212)(162,213)(217,325)(218,326)(219,327)(220,331)(221,332)
(222,333)(223,328)(224,329)(225,330)(226,334)(227,335)(228,336)(229,340)
(230,341)(231,342)(232,337)(233,338)(234,339)(235,343)(236,344)(237,345)
(238,349)(239,350)(240,351)(241,346)(242,347)(243,348)(244,352)(245,353)
(246,354)(247,358)(248,359)(249,360)(250,355)(251,356)(252,357)(253,361)
(254,362)(255,363)(256,367)(257,368)(258,369)(259,364)(260,365)(261,366)
(262,370)(263,371)(264,372)(265,376)(266,377)(267,378)(268,373)(269,374)
(270,375)(271,406)(272,407)(273,408)(274,412)(275,413)(276,414)(277,409)
(278,410)(279,411)(280,415)(281,416)(282,417)(283,421)(284,422)(285,423)
(286,418)(287,419)(288,420)(289,424)(290,425)(291,426)(292,430)(293,431)
(294,432)(295,427)(296,428)(297,429)(298,379)(299,380)(300,381)(301,385)
(302,386)(303,387)(304,382)(305,383)(306,384)(307,388)(308,389)(309,390)
(310,394)(311,395)(312,396)(313,391)(314,392)(315,393)(316,397)(317,398)
(318,399)(319,403)(320,404)(321,405)(322,400)(323,401)(324,402);;
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*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 10, 21)( 11, 20)( 12, 19)
( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 34)
( 32, 36)( 33, 35)( 37, 48)( 38, 47)( 39, 46)( 40, 54)( 41, 53)( 42, 52)
( 43, 51)( 44, 50)( 45, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 75)
( 65, 74)( 66, 73)( 67, 81)( 68, 80)( 69, 79)( 70, 78)( 71, 77)( 72, 76)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 91,102)( 92,101)( 93,100)( 94,108)
( 95,107)( 96,106)( 97,105)( 98,104)( 99,103)(110,111)(112,115)(113,117)
(114,116)(118,129)(119,128)(120,127)(121,135)(122,134)(123,133)(124,132)
(125,131)(126,130)(137,138)(139,142)(140,144)(141,143)(145,156)(146,155)
(147,154)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(164,165)
(166,169)(167,171)(168,170)(172,183)(173,182)(174,181)(175,189)(176,188)
(177,187)(178,186)(179,185)(180,184)(191,192)(193,196)(194,198)(195,197)
(199,210)(200,209)(201,208)(202,216)(203,215)(204,214)(205,213)(206,212)
(207,211)(218,219)(220,223)(221,225)(222,224)(226,237)(227,236)(228,235)
(229,243)(230,242)(231,241)(232,240)(233,239)(234,238)(245,246)(247,250)
(248,252)(249,251)(253,264)(254,263)(255,262)(256,270)(257,269)(258,268)
(259,267)(260,266)(261,265)(272,273)(274,277)(275,279)(276,278)(280,291)
(281,290)(282,289)(283,297)(284,296)(285,295)(286,294)(287,293)(288,292)
(299,300)(301,304)(302,306)(303,305)(307,318)(308,317)(309,316)(310,324)
(311,323)(312,322)(313,321)(314,320)(315,319)(326,327)(328,331)(329,333)
(330,332)(334,345)(335,344)(336,343)(337,351)(338,350)(339,349)(340,348)
(341,347)(342,346)(353,354)(355,358)(356,360)(357,359)(361,372)(362,371)
(363,370)(364,378)(365,377)(366,376)(367,375)(368,374)(369,373)(380,381)
(382,385)(383,387)(384,386)(388,399)(389,398)(390,397)(391,405)(392,404)
(393,403)(394,402)(395,401)(396,400)(407,408)(409,412)(410,414)(411,413)
(415,426)(416,425)(417,424)(418,432)(419,431)(420,430)(421,429)(422,428)
(423,427);
s1 := Sym(432)!( 1,229)( 2,231)( 3,230)( 4,226)( 5,228)( 6,227)( 7,232)
( 8,234)( 9,233)( 10,220)( 11,222)( 12,221)( 13,217)( 14,219)( 15,218)
( 16,223)( 17,225)( 18,224)( 19,240)( 20,239)( 21,238)( 22,237)( 23,236)
( 24,235)( 25,243)( 26,242)( 27,241)( 28,256)( 29,258)( 30,257)( 31,253)
( 32,255)( 33,254)( 34,259)( 35,261)( 36,260)( 37,247)( 38,249)( 39,248)
( 40,244)( 41,246)( 42,245)( 43,250)( 44,252)( 45,251)( 46,267)( 47,266)
( 48,265)( 49,264)( 50,263)( 51,262)( 52,270)( 53,269)( 54,268)( 55,310)
( 56,312)( 57,311)( 58,307)( 59,309)( 60,308)( 61,313)( 62,315)( 63,314)
( 64,301)( 65,303)( 66,302)( 67,298)( 68,300)( 69,299)( 70,304)( 71,306)
( 72,305)( 73,321)( 74,320)( 75,319)( 76,318)( 77,317)( 78,316)( 79,324)
( 80,323)( 81,322)( 82,283)( 83,285)( 84,284)( 85,280)( 86,282)( 87,281)
( 88,286)( 89,288)( 90,287)( 91,274)( 92,276)( 93,275)( 94,271)( 95,273)
( 96,272)( 97,277)( 98,279)( 99,278)(100,294)(101,293)(102,292)(103,291)
(104,290)(105,289)(106,297)(107,296)(108,295)(109,391)(110,393)(111,392)
(112,388)(113,390)(114,389)(115,394)(116,396)(117,395)(118,382)(119,384)
(120,383)(121,379)(122,381)(123,380)(124,385)(125,387)(126,386)(127,402)
(128,401)(129,400)(130,399)(131,398)(132,397)(133,405)(134,404)(135,403)
(136,418)(137,420)(138,419)(139,415)(140,417)(141,416)(142,421)(143,423)
(144,422)(145,409)(146,411)(147,410)(148,406)(149,408)(150,407)(151,412)
(152,414)(153,413)(154,429)(155,428)(156,427)(157,426)(158,425)(159,424)
(160,432)(161,431)(162,430)(163,337)(164,339)(165,338)(166,334)(167,336)
(168,335)(169,340)(170,342)(171,341)(172,328)(173,330)(174,329)(175,325)
(176,327)(177,326)(178,331)(179,333)(180,332)(181,348)(182,347)(183,346)
(184,345)(185,344)(186,343)(187,351)(188,350)(189,349)(190,364)(191,366)
(192,365)(193,361)(194,363)(195,362)(196,367)(197,369)(198,368)(199,355)
(200,357)(201,356)(202,352)(203,354)(204,353)(205,358)(206,360)(207,359)
(208,375)(209,374)(210,373)(211,372)(212,371)(213,370)(214,378)(215,377)
(216,376);
s2 := Sym(432)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)
( 60, 90)( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)
( 68, 98)( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)
( 76,106)( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(109,163)(110,164)
(111,165)(112,169)(113,170)(114,171)(115,166)(116,167)(117,168)(118,172)
(119,173)(120,174)(121,178)(122,179)(123,180)(124,175)(125,176)(126,177)
(127,181)(128,182)(129,183)(130,187)(131,188)(132,189)(133,184)(134,185)
(135,186)(136,190)(137,191)(138,192)(139,196)(140,197)(141,198)(142,193)
(143,194)(144,195)(145,199)(146,200)(147,201)(148,205)(149,206)(150,207)
(151,202)(152,203)(153,204)(154,208)(155,209)(156,210)(157,214)(158,215)
(159,216)(160,211)(161,212)(162,213)(217,325)(218,326)(219,327)(220,331)
(221,332)(222,333)(223,328)(224,329)(225,330)(226,334)(227,335)(228,336)
(229,340)(230,341)(231,342)(232,337)(233,338)(234,339)(235,343)(236,344)
(237,345)(238,349)(239,350)(240,351)(241,346)(242,347)(243,348)(244,352)
(245,353)(246,354)(247,358)(248,359)(249,360)(250,355)(251,356)(252,357)
(253,361)(254,362)(255,363)(256,367)(257,368)(258,369)(259,364)(260,365)
(261,366)(262,370)(263,371)(264,372)(265,376)(266,377)(267,378)(268,373)
(269,374)(270,375)(271,406)(272,407)(273,408)(274,412)(275,413)(276,414)
(277,409)(278,410)(279,411)(280,415)(281,416)(282,417)(283,421)(284,422)
(285,423)(286,418)(287,419)(288,420)(289,424)(290,425)(291,426)(292,430)
(293,431)(294,432)(295,427)(296,428)(297,429)(298,379)(299,380)(300,381)
(301,385)(302,386)(303,387)(304,382)(305,383)(306,384)(307,388)(308,389)
(309,390)(310,394)(311,395)(312,396)(313,391)(314,392)(315,393)(316,397)
(317,398)(318,399)(319,403)(320,404)(321,405)(322,400)(323,401)(324,402);
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*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope