Polytope of Type {8,108}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,108}*1728b
if this polytope has a name.
Group : SmallGroup(1728,390)
Rank : 3
Schlafli Type : {8,108}
Number of vertices, edges, etc : 8, 432, 108
Order of s0s1s2 : 216
Order of s0s1s2s1 : 4
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
3-fold quotients : {8,36}*576b
4-fold quotients : {2,108}*432, {4,54}*432a
6-fold quotients : {4,36}*288a
8-fold quotients : {2,54}*216
9-fold quotients : {8,12}*192b
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 : {8,4}*64b
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.
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)(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)(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)(109,352)(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)(109,145)(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, s2*s0*s1*s0*s1*s2*s0*s1*s0*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)!( 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)(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)(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)(109,352)(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)(109,145)(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, s2*s0*s1*s0*s1*s2*s0*s1*s0*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