Polytope of Type {4,108}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,108}*1728a
if this polytope has a name.
Group : SmallGroup(1728,309)
Rank : 3
Schlafli Type : {4,108}
Number of vertices, edges, etc : 8, 432, 216
Order of s0s1s2 : 108
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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) :
   2-fold quotients : {4,108}*864a
   3-fold quotients : {4,36}*576a
   4-fold quotients : {2,108}*432, {4,54}*432a
   6-fold quotients : {4,36}*288a
   8-fold quotients : {2,54}*216
   9-fold quotients : {4,12}*192a
   12-fold quotients : {2,36}*144, {4,18}*144a
   16-fold quotients : {2,27}*108
   18-fold quotients : {4,12}*96a
   24-fold quotients : {2,18}*72
   27-fold quotients : {4,4}*64
   36-fold quotients : {2,12}*48, {4,6}*48a
   48-fold quotients : {2,9}*36
   54-fold quotients : {4,4}*32
   72-fold quotients : {2,6}*24
   108-fold quotients : {2,4}*16, {4,2}*16
   144-fold quotients : {2,3}*12
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1> of order 2.
      162 facets:
         108 of {2}*4
         54 of {4}*8
      4 vertex figures:
         4 of {108}*216
   P/N, where N=<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*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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      108 facets:
         108 of {4}*8
      6 vertex figures:
         4 of {54}*108
         2 of {108}*216

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

Twisty Puzzle