Polytope of Type {8,108}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,108}*1728a
Also Known As : {8,108|2}. if this polytope has another name.
Group : SmallGroup(1728,342)
Rank : 3
Schlafli Type : {8,108}
Number of vertices, edges, etc : 8, 432, 108
Order of s0s1s2 : 216
Order of s0s1s2s1 : 2
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 : {4,108}*864a, {8,54}*864
   3-fold quotients : {8,36}*576a
   4-fold quotients : {2,108}*432, {4,54}*432a
   6-fold quotients : {4,36}*288a, {8,18}*288
   8-fold quotients : {2,54}*216
   9-fold quotients : {8,12}*192a
   12-fold quotients : {2,36}*144, {4,18}*144a
   16-fold quotients : {2,27}*108
   18-fold quotients : {4,12}*96a, {8,6}*96
   24-fold quotients : {2,18}*72
   27-fold quotients : {8,4}*64a
   36-fold quotients : {2,12}*48, {4,6}*48a
   48-fold quotients : {2,9}*36
   54-fold quotients : {4,4}*32, {8,2}*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):
   None.

Permutation Representation (GAP) : 
s0 := (109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(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)(271,379)(272,380)(273,381)(274,382)(275,383)(276,384)(277,385)(278,386)(279,387)(280,388)(281,389)(282,390)(283,391)(284,392)(285,393)(286,394)(287,395)(288,396)(289,397)(290,398)(291,399)(292,400)(293,401)(294,402)(295,403)(296,404)(297,405)(298,406)(299,407)(300,408)(301,409)(302,410)(303,411)(304,412)(305,413)(306,414)(307,415)(308,416)(309,417)(310,418)(311,419)(312,420)(313,421)(314,422)(315,423)(316,424)(317,425)(318,426)(319,427)(320,428)(321,429)(322,430)(323,431)(324,432);;
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)(109,352)(110,354)(111,353)(112,360)(113,359)(114,358)(115,357)(116,356)(117,355)(118,378)(119,377)(120,376)(121,375)(122,374)(123,373)(124,372)(125,371)(126,370)(127,369)(128,368)(129,367)(130,366)(131,365)(132,364)(133,363)(134,362)(135,361)(136,325)(137,327)(138,326)(139,333)(140,332)(141,331)(142,330)(143,329)(144,328)(145,351)(146,350)(147,349)(148,348)(149,347)(150,346)(151,345)(152,344)(153,343)(154,342)(155,341)(156,340)(157,339)(158,338)(159,337)(160,336)(161,335)(162,334)(163,406)(164,408)(165,407)(166,414)(167,413)(168,412)(169,411)(170,410)(171,409)(172,432)(173,431)(174,430)(175,429)(176,428)(177,427)(178,426)(179,425)(180,424)(181,423)(182,422)(183,421)(184,420)(185,419)(186,418)(187,417)(188,416)(189,415)(190,379)(191,381)(192,380)(193,387)(194,386)(195,385)(196,384)(197,383)(198,382)(199,405)(200,404)(201,403)(202,402)(203,401)(204,400)(205,399)(206,398)(207,397)(208,396)(209,395)(210,394)(211,393)(212,392)(213,391)(214,390)(215,389)(216,388);;
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)(109,118)(110,120)(111,119)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(127,135)(128,134)(129,133)(130,132)(136,145)(137,147)(138,146)(139,153)(140,152)(141,151)(142,150)(143,149)(144,148)(154,162)(155,161)(156,160)(157,159)(163,172)(164,174)(165,173)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(181,189)(182,188)(183,187)(184,186)(190,199)(191,201)(192,200)(193,207)(194,206)(195,205)(196,204)(197,203)(198,202)(208,216)(209,215)(210,214)(211,213)(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)(325,388)(326,390)(327,389)(328,396)(329,395)(330,394)(331,393)(332,392)(333,391)(334,379)(335,381)(336,380)(337,387)(338,386)(339,385)(340,384)(341,383)(342,382)(343,405)(344,404)(345,403)(346,402)(347,401)(348,400)(349,399)(350,398)(351,397)(352,415)(353,417)(354,416)(355,423)(356,422)(357,421)(358,420)(359,419)(360,418)(361,406)(362,408)(363,407)(364,414)(365,413)(366,412)(367,411)(368,410)(369,409)(370,432)(371,431)(372,430)(373,429)(374,428)(375,427)(376,426)(377,425)(378,424);;
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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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)!(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(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)(271,379)(272,380)(273,381)(274,382)(275,383)(276,384)(277,385)(278,386)(279,387)(280,388)(281,389)(282,390)(283,391)(284,392)(285,393)(286,394)(287,395)(288,396)(289,397)(290,398)(291,399)(292,400)(293,401)(294,402)(295,403)(296,404)(297,405)(298,406)(299,407)(300,408)(301,409)(302,410)(303,411)(304,412)(305,413)(306,414)(307,415)(308,416)(309,417)(310,418)(311,419)(312,420)(313,421)(314,422)(315,423)(316,424)(317,425)(318,426)(319,427)(320,428)(321,429)(322,430)(323,431)(324,432);
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)(109,352)(110,354)(111,353)(112,360)(113,359)(114,358)(115,357)(116,356)(117,355)(118,378)(119,377)(120,376)(121,375)(122,374)(123,373)(124,372)(125,371)(126,370)(127,369)(128,368)(129,367)(130,366)(131,365)(132,364)(133,363)(134,362)(135,361)(136,325)(137,327)(138,326)(139,333)(140,332)(141,331)(142,330)(143,329)(144,328)(145,351)(146,350)(147,349)(148,348)(149,347)(150,346)(151,345)(152,344)(153,343)(154,342)(155,341)(156,340)(157,339)(158,338)(159,337)(160,336)(161,335)(162,334)(163,406)(164,408)(165,407)(166,414)(167,413)(168,412)(169,411)(170,410)(171,409)(172,432)(173,431)(174,430)(175,429)(176,428)(177,427)(178,426)(179,425)(180,424)(181,423)(182,422)(183,421)(184,420)(185,419)(186,418)(187,417)(188,416)(189,415)(190,379)(191,381)(192,380)(193,387)(194,386)(195,385)(196,384)(197,383)(198,382)(199,405)(200,404)(201,403)(202,402)(203,401)(204,400)(205,399)(206,398)(207,397)(208,396)(209,395)(210,394)(211,393)(212,392)(213,391)(214,390)(215,389)(216,388);
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)(109,118)(110,120)(111,119)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(127,135)(128,134)(129,133)(130,132)(136,145)(137,147)(138,146)(139,153)(140,152)(141,151)(142,150)(143,149)(144,148)(154,162)(155,161)(156,160)(157,159)(163,172)(164,174)(165,173)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(181,189)(182,188)(183,187)(184,186)(190,199)(191,201)(192,200)(193,207)(194,206)(195,205)(196,204)(197,203)(198,202)(208,216)(209,215)(210,214)(211,213)(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)(325,388)(326,390)(327,389)(328,396)(329,395)(330,394)(331,393)(332,392)(333,391)(334,379)(335,381)(336,380)(337,387)(338,386)(339,385)(340,384)(341,383)(342,382)(343,405)(344,404)(345,403)(346,402)(347,401)(348,400)(349,399)(350,398)(351,397)(352,415)(353,417)(354,416)(355,423)(356,422)(357,421)(358,420)(359,419)(360,418)(361,406)(362,408)(363,407)(364,414)(365,413)(366,412)(367,411)(368,410)(369,409)(370,432)(371,431)(372,430)(373,429)(374,428)(375,427)(376,426)(377,425)(378,424);
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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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