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