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Polytope of Type {8,108}

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.
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