Polytope of Type {6,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,24}*1728d
if this polytope has a name.
Group : SmallGroup(1728,30272)
Rank : 3
Schlafli Type : {6,24}
Number of vertices, edges, etc : 36, 432, 144
Order of s₀s₁s₂ : 6
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 : {6,12}*864a
   3-fold quotients : {6,24}*576d
   4-fold quotients : {3,12}*432
   6-fold quotients : {6,12}*288b
   8-fold quotients : {6,6}*216c
   9-fold quotients : {6,8}*192c
   12-fold quotients : {3,12}*144
   16-fold quotients : {3,6}*108
   18-fold quotients : {6,4}*96
   24-fold quotients : {6,6}*72c
   36-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   48-fold quotients : {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)
(  9,225)( 10,226)( 11,228)( 12,227)( 13,231)( 14,232)( 15,229)( 16,230)
( 17,233)( 18,234)( 19,236)( 20,235)( 21,239)( 22,240)( 23,237)( 24,238)
( 25,265)( 26,266)( 27,268)( 28,267)( 29,271)( 30,272)( 31,269)( 32,270)
( 33,273)( 34,274)( 35,276)( 36,275)( 37,279)( 38,280)( 39,277)( 40,278)
( 41,281)( 42,282)( 43,284)( 44,283)( 45,287)( 46,288)( 47,285)( 48,286)
( 49,241)( 50,242)( 51,244)( 52,243)( 53,247)( 54,248)( 55,245)( 56,246)
( 57,249)( 58,250)( 59,252)( 60,251)( 61,255)( 62,256)( 63,253)( 64,254)
( 65,257)( 66,258)( 67,260)( 68,259)( 69,263)( 70,264)( 71,261)( 72,262)
( 73,361)( 74,362)( 75,364)( 76,363)( 77,367)( 78,368)( 79,365)( 80,366)
( 81,369)( 82,370)( 83,372)( 84,371)( 85,375)( 86,376)( 87,373)( 88,374)
( 89,377)( 90,378)( 91,380)( 92,379)( 93,383)( 94,384)( 95,381)( 96,382)
( 97,409)( 98,410)( 99,412)(100,411)(101,415)(102,416)(103,413)(104,414)
(105,417)(106,418)(107,420)(108,419)(109,423)(110,424)(111,421)(112,422)
(113,425)(114,426)(115,428)(116,427)(117,431)(118,432)(119,429)(120,430)
(121,385)(122,386)(123,388)(124,387)(125,391)(126,392)(127,389)(128,390)
(129,393)(130,394)(131,396)(132,395)(133,399)(134,400)(135,397)(136,398)
(137,401)(138,402)(139,404)(140,403)(141,407)(142,408)(143,405)(144,406)
(145,289)(146,290)(147,292)(148,291)(149,295)(150,296)(151,293)(152,294)
(153,297)(154,298)(155,300)(156,299)(157,303)(158,304)(159,301)(160,302)
(161,305)(162,306)(163,308)(164,307)(165,311)(166,312)(167,309)(168,310)
(169,337)(170,338)(171,340)(172,339)(173,343)(174,344)(175,341)(176,342)
(177,345)(178,346)(179,348)(180,347)(181,351)(182,352)(183,349)(184,350)
(185,353)(186,354)(187,356)(188,355)(189,359)(190,360)(191,357)(192,358)
(193,313)(194,314)(195,316)(196,315)(197,319)(198,320)(199,317)(200,318)
(201,321)(202,322)(203,324)(204,323)(205,327)(206,328)(207,325)(208,326)
(209,329)(210,330)(211,332)(212,331)(213,335)(214,336)(215,333)(216,334);;
s1 := (  1,177)(  2,178)(  3,182)(  4,181)(  5,180)(  6,179)(  7,184)(  8,183)
(  9,185)( 10,186)( 11,190)( 12,189)( 13,188)( 14,187)( 15,192)( 16,191)
( 17,169)( 18,170)( 19,174)( 20,173)( 21,172)( 22,171)( 23,176)( 24,175)
( 25,161)( 26,162)( 27,166)( 28,165)( 29,164)( 30,163)( 31,168)( 32,167)
( 33,145)( 34,146)( 35,150)( 36,149)( 37,148)( 38,147)( 39,152)( 40,151)
( 41,153)( 42,154)( 43,158)( 44,157)( 45,156)( 46,155)( 47,160)( 48,159)
( 49,193)( 50,194)( 51,198)( 52,197)( 53,196)( 54,195)( 55,200)( 56,199)
( 57,201)( 58,202)( 59,206)( 60,205)( 61,204)( 62,203)( 63,208)( 64,207)
( 65,209)( 66,210)( 67,214)( 68,213)( 69,212)( 70,211)( 71,216)( 72,215)
( 73,105)( 74,106)( 75,110)( 76,109)( 77,108)( 78,107)( 79,112)( 80,111)
( 81,113)( 82,114)( 83,118)( 84,117)( 85,116)( 86,115)( 87,120)( 88,119)
( 89, 97)( 90, 98)( 91,102)( 92,101)( 93,100)( 94, 99)( 95,104)( 96,103)
(123,126)(124,125)(127,128)(131,134)(132,133)(135,136)(139,142)(140,141)
(143,144)(217,393)(218,394)(219,398)(220,397)(221,396)(222,395)(223,400)
(224,399)(225,401)(226,402)(227,406)(228,405)(229,404)(230,403)(231,408)
(232,407)(233,385)(234,386)(235,390)(236,389)(237,388)(238,387)(239,392)
(240,391)(241,377)(242,378)(243,382)(244,381)(245,380)(246,379)(247,384)
(248,383)(249,361)(250,362)(251,366)(252,365)(253,364)(254,363)(255,368)
(256,367)(257,369)(258,370)(259,374)(260,373)(261,372)(262,371)(263,376)
(264,375)(265,409)(266,410)(267,414)(268,413)(269,412)(270,411)(271,416)
(272,415)(273,417)(274,418)(275,422)(276,421)(277,420)(278,419)(279,424)
(280,423)(281,425)(282,426)(283,430)(284,429)(285,428)(286,427)(287,432)
(288,431)(289,321)(290,322)(291,326)(292,325)(293,324)(294,323)(295,328)
(296,327)(297,329)(298,330)(299,334)(300,333)(301,332)(302,331)(303,336)
(304,335)(305,313)(306,314)(307,318)(308,317)(309,316)(310,315)(311,320)
(312,319)(339,342)(340,341)(343,344)(347,350)(348,349)(351,352)(355,358)
(356,357)(359,360);;
s2 := (  1,219)(  2,220)(  3,218)(  4,217)(  5,223)(  6,224)(  7,222)(  8,221)
(  9,235)( 10,236)( 11,234)( 12,233)( 13,239)( 14,240)( 15,238)( 16,237)
( 17,227)( 18,228)( 19,226)( 20,225)( 21,231)( 22,232)( 23,230)( 24,229)
( 25,267)( 26,268)( 27,266)( 28,265)( 29,271)( 30,272)( 31,270)( 32,269)
( 33,283)( 34,284)( 35,282)( 36,281)( 37,287)( 38,288)( 39,286)( 40,285)
( 41,275)( 42,276)( 43,274)( 44,273)( 45,279)( 46,280)( 47,278)( 48,277)
( 49,243)( 50,244)( 51,242)( 52,241)( 53,247)( 54,248)( 55,246)( 56,245)
( 57,259)( 58,260)( 59,258)( 60,257)( 61,263)( 62,264)( 63,262)( 64,261)
( 65,251)( 66,252)( 67,250)( 68,249)( 69,255)( 70,256)( 71,254)( 72,253)
( 73,291)( 74,292)( 75,290)( 76,289)( 77,295)( 78,296)( 79,294)( 80,293)
( 81,307)( 82,308)( 83,306)( 84,305)( 85,311)( 86,312)( 87,310)( 88,309)
( 89,299)( 90,300)( 91,298)( 92,297)( 93,303)( 94,304)( 95,302)( 96,301)
( 97,339)( 98,340)( 99,338)(100,337)(101,343)(102,344)(103,342)(104,341)
(105,355)(106,356)(107,354)(108,353)(109,359)(110,360)(111,358)(112,357)
(113,347)(114,348)(115,346)(116,345)(117,351)(118,352)(119,350)(120,349)
(121,315)(122,316)(123,314)(124,313)(125,319)(126,320)(127,318)(128,317)
(129,331)(130,332)(131,330)(132,329)(133,335)(134,336)(135,334)(136,333)
(137,323)(138,324)(139,322)(140,321)(141,327)(142,328)(143,326)(144,325)
(145,363)(146,364)(147,362)(148,361)(149,367)(150,368)(151,366)(152,365)
(153,379)(154,380)(155,378)(156,377)(157,383)(158,384)(159,382)(160,381)
(161,371)(162,372)(163,370)(164,369)(165,375)(166,376)(167,374)(168,373)
(169,411)(170,412)(171,410)(172,409)(173,415)(174,416)(175,414)(176,413)
(177,427)(178,428)(179,426)(180,425)(181,431)(182,432)(183,430)(184,429)
(185,419)(186,420)(187,418)(188,417)(189,423)(190,424)(191,422)(192,421)
(193,387)(194,388)(195,386)(196,385)(197,391)(198,392)(199,390)(200,389)
(201,403)(202,404)(203,402)(204,401)(205,407)(206,408)(207,406)(208,405)
(209,395)(210,396)(211,394)(212,393)(213,399)(214,400)(215,398)(216,397);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*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)(  9,225)( 10,226)( 11,228)( 12,227)( 13,231)( 14,232)( 15,229)
( 16,230)( 17,233)( 18,234)( 19,236)( 20,235)( 21,239)( 22,240)( 23,237)
( 24,238)( 25,265)( 26,266)( 27,268)( 28,267)( 29,271)( 30,272)( 31,269)
( 32,270)( 33,273)( 34,274)( 35,276)( 36,275)( 37,279)( 38,280)( 39,277)
( 40,278)( 41,281)( 42,282)( 43,284)( 44,283)( 45,287)( 46,288)( 47,285)
( 48,286)( 49,241)( 50,242)( 51,244)( 52,243)( 53,247)( 54,248)( 55,245)
( 56,246)( 57,249)( 58,250)( 59,252)( 60,251)( 61,255)( 62,256)( 63,253)
( 64,254)( 65,257)( 66,258)( 67,260)( 68,259)( 69,263)( 70,264)( 71,261)
( 72,262)( 73,361)( 74,362)( 75,364)( 76,363)( 77,367)( 78,368)( 79,365)
( 80,366)( 81,369)( 82,370)( 83,372)( 84,371)( 85,375)( 86,376)( 87,373)
( 88,374)( 89,377)( 90,378)( 91,380)( 92,379)( 93,383)( 94,384)( 95,381)
( 96,382)( 97,409)( 98,410)( 99,412)(100,411)(101,415)(102,416)(103,413)
(104,414)(105,417)(106,418)(107,420)(108,419)(109,423)(110,424)(111,421)
(112,422)(113,425)(114,426)(115,428)(116,427)(117,431)(118,432)(119,429)
(120,430)(121,385)(122,386)(123,388)(124,387)(125,391)(126,392)(127,389)
(128,390)(129,393)(130,394)(131,396)(132,395)(133,399)(134,400)(135,397)
(136,398)(137,401)(138,402)(139,404)(140,403)(141,407)(142,408)(143,405)
(144,406)(145,289)(146,290)(147,292)(148,291)(149,295)(150,296)(151,293)
(152,294)(153,297)(154,298)(155,300)(156,299)(157,303)(158,304)(159,301)
(160,302)(161,305)(162,306)(163,308)(164,307)(165,311)(166,312)(167,309)
(168,310)(169,337)(170,338)(171,340)(172,339)(173,343)(174,344)(175,341)
(176,342)(177,345)(178,346)(179,348)(180,347)(181,351)(182,352)(183,349)
(184,350)(185,353)(186,354)(187,356)(188,355)(189,359)(190,360)(191,357)
(192,358)(193,313)(194,314)(195,316)(196,315)(197,319)(198,320)(199,317)
(200,318)(201,321)(202,322)(203,324)(204,323)(205,327)(206,328)(207,325)
(208,326)(209,329)(210,330)(211,332)(212,331)(213,335)(214,336)(215,333)
(216,334);
s1 := Sym(432)!(  1,177)(  2,178)(  3,182)(  4,181)(  5,180)(  6,179)(  7,184)
(  8,183)(  9,185)( 10,186)( 11,190)( 12,189)( 13,188)( 14,187)( 15,192)
( 16,191)( 17,169)( 18,170)( 19,174)( 20,173)( 21,172)( 22,171)( 23,176)
( 24,175)( 25,161)( 26,162)( 27,166)( 28,165)( 29,164)( 30,163)( 31,168)
( 32,167)( 33,145)( 34,146)( 35,150)( 36,149)( 37,148)( 38,147)( 39,152)
( 40,151)( 41,153)( 42,154)( 43,158)( 44,157)( 45,156)( 46,155)( 47,160)
( 48,159)( 49,193)( 50,194)( 51,198)( 52,197)( 53,196)( 54,195)( 55,200)
( 56,199)( 57,201)( 58,202)( 59,206)( 60,205)( 61,204)( 62,203)( 63,208)
( 64,207)( 65,209)( 66,210)( 67,214)( 68,213)( 69,212)( 70,211)( 71,216)
( 72,215)( 73,105)( 74,106)( 75,110)( 76,109)( 77,108)( 78,107)( 79,112)
( 80,111)( 81,113)( 82,114)( 83,118)( 84,117)( 85,116)( 86,115)( 87,120)
( 88,119)( 89, 97)( 90, 98)( 91,102)( 92,101)( 93,100)( 94, 99)( 95,104)
( 96,103)(123,126)(124,125)(127,128)(131,134)(132,133)(135,136)(139,142)
(140,141)(143,144)(217,393)(218,394)(219,398)(220,397)(221,396)(222,395)
(223,400)(224,399)(225,401)(226,402)(227,406)(228,405)(229,404)(230,403)
(231,408)(232,407)(233,385)(234,386)(235,390)(236,389)(237,388)(238,387)
(239,392)(240,391)(241,377)(242,378)(243,382)(244,381)(245,380)(246,379)
(247,384)(248,383)(249,361)(250,362)(251,366)(252,365)(253,364)(254,363)
(255,368)(256,367)(257,369)(258,370)(259,374)(260,373)(261,372)(262,371)
(263,376)(264,375)(265,409)(266,410)(267,414)(268,413)(269,412)(270,411)
(271,416)(272,415)(273,417)(274,418)(275,422)(276,421)(277,420)(278,419)
(279,424)(280,423)(281,425)(282,426)(283,430)(284,429)(285,428)(286,427)
(287,432)(288,431)(289,321)(290,322)(291,326)(292,325)(293,324)(294,323)
(295,328)(296,327)(297,329)(298,330)(299,334)(300,333)(301,332)(302,331)
(303,336)(304,335)(305,313)(306,314)(307,318)(308,317)(309,316)(310,315)
(311,320)(312,319)(339,342)(340,341)(343,344)(347,350)(348,349)(351,352)
(355,358)(356,357)(359,360);
s2 := Sym(432)!(  1,219)(  2,220)(  3,218)(  4,217)(  5,223)(  6,224)(  7,222)
(  8,221)(  9,235)( 10,236)( 11,234)( 12,233)( 13,239)( 14,240)( 15,238)
( 16,237)( 17,227)( 18,228)( 19,226)( 20,225)( 21,231)( 22,232)( 23,230)
( 24,229)( 25,267)( 26,268)( 27,266)( 28,265)( 29,271)( 30,272)( 31,270)
( 32,269)( 33,283)( 34,284)( 35,282)( 36,281)( 37,287)( 38,288)( 39,286)
( 40,285)( 41,275)( 42,276)( 43,274)( 44,273)( 45,279)( 46,280)( 47,278)
( 48,277)( 49,243)( 50,244)( 51,242)( 52,241)( 53,247)( 54,248)( 55,246)
( 56,245)( 57,259)( 58,260)( 59,258)( 60,257)( 61,263)( 62,264)( 63,262)
( 64,261)( 65,251)( 66,252)( 67,250)( 68,249)( 69,255)( 70,256)( 71,254)
( 72,253)( 73,291)( 74,292)( 75,290)( 76,289)( 77,295)( 78,296)( 79,294)
( 80,293)( 81,307)( 82,308)( 83,306)( 84,305)( 85,311)( 86,312)( 87,310)
( 88,309)( 89,299)( 90,300)( 91,298)( 92,297)( 93,303)( 94,304)( 95,302)
( 96,301)( 97,339)( 98,340)( 99,338)(100,337)(101,343)(102,344)(103,342)
(104,341)(105,355)(106,356)(107,354)(108,353)(109,359)(110,360)(111,358)
(112,357)(113,347)(114,348)(115,346)(116,345)(117,351)(118,352)(119,350)
(120,349)(121,315)(122,316)(123,314)(124,313)(125,319)(126,320)(127,318)
(128,317)(129,331)(130,332)(131,330)(132,329)(133,335)(134,336)(135,334)
(136,333)(137,323)(138,324)(139,322)(140,321)(141,327)(142,328)(143,326)
(144,325)(145,363)(146,364)(147,362)(148,361)(149,367)(150,368)(151,366)
(152,365)(153,379)(154,380)(155,378)(156,377)(157,383)(158,384)(159,382)
(160,381)(161,371)(162,372)(163,370)(164,369)(165,375)(166,376)(167,374)
(168,373)(169,411)(170,412)(171,410)(172,409)(173,415)(174,416)(175,414)
(176,413)(177,427)(178,428)(179,426)(180,425)(181,431)(182,432)(183,430)
(184,429)(185,419)(186,420)(187,418)(188,417)(189,423)(190,424)(191,422)
(192,421)(193,387)(194,388)(195,386)(196,385)(197,391)(198,392)(199,390)
(200,389)(201,403)(202,404)(203,402)(204,401)(205,407)(206,408)(207,406)
(208,405)(209,395)(210,396)(211,394)(212,393)(213,399)(214,400)(215,398)
(216,397);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1 >;

References : None.
to this polytope