Polytope of Type {18,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,24}*1728d
if this polytope has a name.
Group : SmallGroup(1728,30258)
Rank : 3
Schlafli Type : {18,24}
Number of vertices, edges, etc : 36, 432, 48
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) :
   2-fold quotients : {18,12}*864b
   3-fold quotients : {18,8}*576c, {6,24}*576d
   4-fold quotients : {9,12}*432
   6-fold quotients : {18,4}*288, {6,12}*288b
   8-fold quotients : {18,6}*216b
   9-fold quotients : {6,8}*192c
   12-fold quotients : {9,4}*144, {18,4}*144b, {18,4}*144c, {3,12}*144
   16-fold quotients : {9,6}*108
   18-fold quotients : {6,4}*96
   24-fold quotients : {9,4}*72, {18,2}*72, {6,6}*72c
   36-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   48-fold quotients : {9,2}*36, {3,6}*36
   72-fold quotients : {3,4}*24, {6,2}*24
   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 := (  1,217)(  2,218)(  3,220)(  4,219)(  5,223)(  6,224)(  7,221)(  8,222)
(  9,233)( 10,234)( 11,236)( 12,235)( 13,239)( 14,240)( 15,237)( 16,238)
( 17,225)( 18,226)( 19,228)( 20,227)( 21,231)( 22,232)( 23,229)( 24,230)
( 25,265)( 26,266)( 27,268)( 28,267)( 29,271)( 30,272)( 31,269)( 32,270)
( 33,281)( 34,282)( 35,284)( 36,283)( 37,287)( 38,288)( 39,285)( 40,286)
( 41,273)( 42,274)( 43,276)( 44,275)( 45,279)( 46,280)( 47,277)( 48,278)
( 49,241)( 50,242)( 51,244)( 52,243)( 53,247)( 54,248)( 55,245)( 56,246)
( 57,257)( 58,258)( 59,260)( 60,259)( 61,263)( 62,264)( 63,261)( 64,262)
( 65,249)( 66,250)( 67,252)( 68,251)( 69,255)( 70,256)( 71,253)( 72,254)
( 73,377)( 74,378)( 75,380)( 76,379)( 77,383)( 78,384)( 79,381)( 80,382)
( 81,369)( 82,370)( 83,372)( 84,371)( 85,375)( 86,376)( 87,373)( 88,374)
( 89,361)( 90,362)( 91,364)( 92,363)( 93,367)( 94,368)( 95,365)( 96,366)
( 97,425)( 98,426)( 99,428)(100,427)(101,431)(102,432)(103,429)(104,430)
(105,417)(106,418)(107,420)(108,419)(109,423)(110,424)(111,421)(112,422)
(113,409)(114,410)(115,412)(116,411)(117,415)(118,416)(119,413)(120,414)
(121,401)(122,402)(123,404)(124,403)(125,407)(126,408)(127,405)(128,406)
(129,393)(130,394)(131,396)(132,395)(133,399)(134,400)(135,397)(136,398)
(137,385)(138,386)(139,388)(140,387)(141,391)(142,392)(143,389)(144,390)
(145,305)(146,306)(147,308)(148,307)(149,311)(150,312)(151,309)(152,310)
(153,297)(154,298)(155,300)(156,299)(157,303)(158,304)(159,301)(160,302)
(161,289)(162,290)(163,292)(164,291)(165,295)(166,296)(167,293)(168,294)
(169,353)(170,354)(171,356)(172,355)(173,359)(174,360)(175,357)(176,358)
(177,345)(178,346)(179,348)(180,347)(181,351)(182,352)(183,349)(184,350)
(185,337)(186,338)(187,340)(188,339)(189,343)(190,344)(191,341)(192,342)
(193,329)(194,330)(195,332)(196,331)(197,335)(198,336)(199,333)(200,334)
(201,321)(202,322)(203,324)(204,323)(205,327)(206,328)(207,325)(208,326)
(209,313)(210,314)(211,316)(212,315)(213,319)(214,320)(215,317)(216,318);;
s1 := (  1, 97)(  2, 98)(  3,103)(  4,104)(  5,102)(  6,101)(  7, 99)(  8,100)
(  9,113)( 10,114)( 11,119)( 12,120)( 13,118)( 14,117)( 15,115)( 16,116)
( 17,105)( 18,106)( 19,111)( 20,112)( 21,110)( 22,109)( 23,107)( 24,108)
( 25, 73)( 26, 74)( 27, 79)( 28, 80)( 29, 78)( 30, 77)( 31, 75)( 32, 76)
( 33, 89)( 34, 90)( 35, 95)( 36, 96)( 37, 94)( 38, 93)( 39, 91)( 40, 92)
( 41, 81)( 42, 82)( 43, 87)( 44, 88)( 45, 86)( 46, 85)( 47, 83)( 48, 84)
( 49,121)( 50,122)( 51,127)( 52,128)( 53,126)( 54,125)( 55,123)( 56,124)
( 57,137)( 58,138)( 59,143)( 60,144)( 61,142)( 62,141)( 63,139)( 64,140)
( 65,129)( 66,130)( 67,135)( 68,136)( 69,134)( 70,133)( 71,131)( 72,132)
(145,185)(146,186)(147,191)(148,192)(149,190)(150,189)(151,187)(152,188)
(153,177)(154,178)(155,183)(156,184)(157,182)(158,181)(159,179)(160,180)
(161,169)(162,170)(163,175)(164,176)(165,174)(166,173)(167,171)(168,172)
(193,209)(194,210)(195,215)(196,216)(197,214)(198,213)(199,211)(200,212)
(203,207)(204,208)(205,206)(217,313)(218,314)(219,319)(220,320)(221,318)
(222,317)(223,315)(224,316)(225,329)(226,330)(227,335)(228,336)(229,334)
(230,333)(231,331)(232,332)(233,321)(234,322)(235,327)(236,328)(237,326)
(238,325)(239,323)(240,324)(241,289)(242,290)(243,295)(244,296)(245,294)
(246,293)(247,291)(248,292)(249,305)(250,306)(251,311)(252,312)(253,310)
(254,309)(255,307)(256,308)(257,297)(258,298)(259,303)(260,304)(261,302)
(262,301)(263,299)(264,300)(265,337)(266,338)(267,343)(268,344)(269,342)
(270,341)(271,339)(272,340)(273,353)(274,354)(275,359)(276,360)(277,358)
(278,357)(279,355)(280,356)(281,345)(282,346)(283,351)(284,352)(285,350)
(286,349)(287,347)(288,348)(361,401)(362,402)(363,407)(364,408)(365,406)
(366,405)(367,403)(368,404)(369,393)(370,394)(371,399)(372,400)(373,398)
(374,397)(375,395)(376,396)(377,385)(378,386)(379,391)(380,392)(381,390)
(382,389)(383,387)(384,388)(409,425)(410,426)(411,431)(412,432)(413,430)
(414,429)(415,427)(416,428)(419,423)(420,424)(421,422);;
s2 := (  1,219)(  2,220)(  3,218)(  4,217)(  5,223)(  6,224)(  7,222)(  8,221)
(  9,227)( 10,228)( 11,226)( 12,225)( 13,231)( 14,232)( 15,230)( 16,229)
( 17,235)( 18,236)( 19,234)( 20,233)( 21,239)( 22,240)( 23,238)( 24,237)
( 25,267)( 26,268)( 27,266)( 28,265)( 29,271)( 30,272)( 31,270)( 32,269)
( 33,275)( 34,276)( 35,274)( 36,273)( 37,279)( 38,280)( 39,278)( 40,277)
( 41,283)( 42,284)( 43,282)( 44,281)( 45,287)( 46,288)( 47,286)( 48,285)
( 49,243)( 50,244)( 51,242)( 52,241)( 53,247)( 54,248)( 55,246)( 56,245)
( 57,251)( 58,252)( 59,250)( 60,249)( 61,255)( 62,256)( 63,254)( 64,253)
( 65,259)( 66,260)( 67,258)( 68,257)( 69,263)( 70,264)( 71,262)( 72,261)
( 73,291)( 74,292)( 75,290)( 76,289)( 77,295)( 78,296)( 79,294)( 80,293)
( 81,299)( 82,300)( 83,298)( 84,297)( 85,303)( 86,304)( 87,302)( 88,301)
( 89,307)( 90,308)( 91,306)( 92,305)( 93,311)( 94,312)( 95,310)( 96,309)
( 97,339)( 98,340)( 99,338)(100,337)(101,343)(102,344)(103,342)(104,341)
(105,347)(106,348)(107,346)(108,345)(109,351)(110,352)(111,350)(112,349)
(113,355)(114,356)(115,354)(116,353)(117,359)(118,360)(119,358)(120,357)
(121,315)(122,316)(123,314)(124,313)(125,319)(126,320)(127,318)(128,317)
(129,323)(130,324)(131,322)(132,321)(133,327)(134,328)(135,326)(136,325)
(137,331)(138,332)(139,330)(140,329)(141,335)(142,336)(143,334)(144,333)
(145,363)(146,364)(147,362)(148,361)(149,367)(150,368)(151,366)(152,365)
(153,371)(154,372)(155,370)(156,369)(157,375)(158,376)(159,374)(160,373)
(161,379)(162,380)(163,378)(164,377)(165,383)(166,384)(167,382)(168,381)
(169,411)(170,412)(171,410)(172,409)(173,415)(174,416)(175,414)(176,413)
(177,419)(178,420)(179,418)(180,417)(181,423)(182,424)(183,422)(184,421)
(185,427)(186,428)(187,426)(188,425)(189,431)(190,432)(191,430)(192,429)
(193,387)(194,388)(195,386)(196,385)(197,391)(198,392)(199,390)(200,389)
(201,395)(202,396)(203,394)(204,393)(205,399)(206,400)(207,398)(208,397)
(209,403)(210,404)(211,402)(212,401)(213,407)(214,408)(215,406)(216,405);;
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*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*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)!(  1,217)(  2,218)(  3,220)(  4,219)(  5,223)(  6,224)(  7,221)
(  8,222)(  9,233)( 10,234)( 11,236)( 12,235)( 13,239)( 14,240)( 15,237)
( 16,238)( 17,225)( 18,226)( 19,228)( 20,227)( 21,231)( 22,232)( 23,229)
( 24,230)( 25,265)( 26,266)( 27,268)( 28,267)( 29,271)( 30,272)( 31,269)
( 32,270)( 33,281)( 34,282)( 35,284)( 36,283)( 37,287)( 38,288)( 39,285)
( 40,286)( 41,273)( 42,274)( 43,276)( 44,275)( 45,279)( 46,280)( 47,277)
( 48,278)( 49,241)( 50,242)( 51,244)( 52,243)( 53,247)( 54,248)( 55,245)
( 56,246)( 57,257)( 58,258)( 59,260)( 60,259)( 61,263)( 62,264)( 63,261)
( 64,262)( 65,249)( 66,250)( 67,252)( 68,251)( 69,255)( 70,256)( 71,253)
( 72,254)( 73,377)( 74,378)( 75,380)( 76,379)( 77,383)( 78,384)( 79,381)
( 80,382)( 81,369)( 82,370)( 83,372)( 84,371)( 85,375)( 86,376)( 87,373)
( 88,374)( 89,361)( 90,362)( 91,364)( 92,363)( 93,367)( 94,368)( 95,365)
( 96,366)( 97,425)( 98,426)( 99,428)(100,427)(101,431)(102,432)(103,429)
(104,430)(105,417)(106,418)(107,420)(108,419)(109,423)(110,424)(111,421)
(112,422)(113,409)(114,410)(115,412)(116,411)(117,415)(118,416)(119,413)
(120,414)(121,401)(122,402)(123,404)(124,403)(125,407)(126,408)(127,405)
(128,406)(129,393)(130,394)(131,396)(132,395)(133,399)(134,400)(135,397)
(136,398)(137,385)(138,386)(139,388)(140,387)(141,391)(142,392)(143,389)
(144,390)(145,305)(146,306)(147,308)(148,307)(149,311)(150,312)(151,309)
(152,310)(153,297)(154,298)(155,300)(156,299)(157,303)(158,304)(159,301)
(160,302)(161,289)(162,290)(163,292)(164,291)(165,295)(166,296)(167,293)
(168,294)(169,353)(170,354)(171,356)(172,355)(173,359)(174,360)(175,357)
(176,358)(177,345)(178,346)(179,348)(180,347)(181,351)(182,352)(183,349)
(184,350)(185,337)(186,338)(187,340)(188,339)(189,343)(190,344)(191,341)
(192,342)(193,329)(194,330)(195,332)(196,331)(197,335)(198,336)(199,333)
(200,334)(201,321)(202,322)(203,324)(204,323)(205,327)(206,328)(207,325)
(208,326)(209,313)(210,314)(211,316)(212,315)(213,319)(214,320)(215,317)
(216,318);
s1 := Sym(432)!(  1, 97)(  2, 98)(  3,103)(  4,104)(  5,102)(  6,101)(  7, 99)
(  8,100)(  9,113)( 10,114)( 11,119)( 12,120)( 13,118)( 14,117)( 15,115)
( 16,116)( 17,105)( 18,106)( 19,111)( 20,112)( 21,110)( 22,109)( 23,107)
( 24,108)( 25, 73)( 26, 74)( 27, 79)( 28, 80)( 29, 78)( 30, 77)( 31, 75)
( 32, 76)( 33, 89)( 34, 90)( 35, 95)( 36, 96)( 37, 94)( 38, 93)( 39, 91)
( 40, 92)( 41, 81)( 42, 82)( 43, 87)( 44, 88)( 45, 86)( 46, 85)( 47, 83)
( 48, 84)( 49,121)( 50,122)( 51,127)( 52,128)( 53,126)( 54,125)( 55,123)
( 56,124)( 57,137)( 58,138)( 59,143)( 60,144)( 61,142)( 62,141)( 63,139)
( 64,140)( 65,129)( 66,130)( 67,135)( 68,136)( 69,134)( 70,133)( 71,131)
( 72,132)(145,185)(146,186)(147,191)(148,192)(149,190)(150,189)(151,187)
(152,188)(153,177)(154,178)(155,183)(156,184)(157,182)(158,181)(159,179)
(160,180)(161,169)(162,170)(163,175)(164,176)(165,174)(166,173)(167,171)
(168,172)(193,209)(194,210)(195,215)(196,216)(197,214)(198,213)(199,211)
(200,212)(203,207)(204,208)(205,206)(217,313)(218,314)(219,319)(220,320)
(221,318)(222,317)(223,315)(224,316)(225,329)(226,330)(227,335)(228,336)
(229,334)(230,333)(231,331)(232,332)(233,321)(234,322)(235,327)(236,328)
(237,326)(238,325)(239,323)(240,324)(241,289)(242,290)(243,295)(244,296)
(245,294)(246,293)(247,291)(248,292)(249,305)(250,306)(251,311)(252,312)
(253,310)(254,309)(255,307)(256,308)(257,297)(258,298)(259,303)(260,304)
(261,302)(262,301)(263,299)(264,300)(265,337)(266,338)(267,343)(268,344)
(269,342)(270,341)(271,339)(272,340)(273,353)(274,354)(275,359)(276,360)
(277,358)(278,357)(279,355)(280,356)(281,345)(282,346)(283,351)(284,352)
(285,350)(286,349)(287,347)(288,348)(361,401)(362,402)(363,407)(364,408)
(365,406)(366,405)(367,403)(368,404)(369,393)(370,394)(371,399)(372,400)
(373,398)(374,397)(375,395)(376,396)(377,385)(378,386)(379,391)(380,392)
(381,390)(382,389)(383,387)(384,388)(409,425)(410,426)(411,431)(412,432)
(413,430)(414,429)(415,427)(416,428)(419,423)(420,424)(421,422);
s2 := Sym(432)!(  1,219)(  2,220)(  3,218)(  4,217)(  5,223)(  6,224)(  7,222)
(  8,221)(  9,227)( 10,228)( 11,226)( 12,225)( 13,231)( 14,232)( 15,230)
( 16,229)( 17,235)( 18,236)( 19,234)( 20,233)( 21,239)( 22,240)( 23,238)
( 24,237)( 25,267)( 26,268)( 27,266)( 28,265)( 29,271)( 30,272)( 31,270)
( 32,269)( 33,275)( 34,276)( 35,274)( 36,273)( 37,279)( 38,280)( 39,278)
( 40,277)( 41,283)( 42,284)( 43,282)( 44,281)( 45,287)( 46,288)( 47,286)
( 48,285)( 49,243)( 50,244)( 51,242)( 52,241)( 53,247)( 54,248)( 55,246)
( 56,245)( 57,251)( 58,252)( 59,250)( 60,249)( 61,255)( 62,256)( 63,254)
( 64,253)( 65,259)( 66,260)( 67,258)( 68,257)( 69,263)( 70,264)( 71,262)
( 72,261)( 73,291)( 74,292)( 75,290)( 76,289)( 77,295)( 78,296)( 79,294)
( 80,293)( 81,299)( 82,300)( 83,298)( 84,297)( 85,303)( 86,304)( 87,302)
( 88,301)( 89,307)( 90,308)( 91,306)( 92,305)( 93,311)( 94,312)( 95,310)
( 96,309)( 97,339)( 98,340)( 99,338)(100,337)(101,343)(102,344)(103,342)
(104,341)(105,347)(106,348)(107,346)(108,345)(109,351)(110,352)(111,350)
(112,349)(113,355)(114,356)(115,354)(116,353)(117,359)(118,360)(119,358)
(120,357)(121,315)(122,316)(123,314)(124,313)(125,319)(126,320)(127,318)
(128,317)(129,323)(130,324)(131,322)(132,321)(133,327)(134,328)(135,326)
(136,325)(137,331)(138,332)(139,330)(140,329)(141,335)(142,336)(143,334)
(144,333)(145,363)(146,364)(147,362)(148,361)(149,367)(150,368)(151,366)
(152,365)(153,371)(154,372)(155,370)(156,369)(157,375)(158,376)(159,374)
(160,373)(161,379)(162,380)(163,378)(164,377)(165,383)(166,384)(167,382)
(168,381)(169,411)(170,412)(171,410)(172,409)(173,415)(174,416)(175,414)
(176,413)(177,419)(178,420)(179,418)(180,417)(181,423)(182,424)(183,422)
(184,421)(185,427)(186,428)(187,426)(188,425)(189,431)(190,432)(191,430)
(192,429)(193,387)(194,388)(195,386)(196,385)(197,391)(198,392)(199,390)
(200,389)(201,395)(202,396)(203,394)(204,393)(205,399)(206,400)(207,398)
(208,397)(209,403)(210,404)(211,402)(212,401)(213,407)(214,408)(215,406)
(216,405);
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*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*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