Polytope of Type {4,108}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,108}*864b
if this polytope has a name.
Group : SmallGroup(864,613)
Rank : 3
Schlafli Type : {4,108}
Number of vertices, edges, etc : 4, 216, 108
Order of s0s1s2 : 108
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,108,2} of size 1728
Vertex Figure Of :
{2,4,108} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,54}*432b
3-fold quotients : {4,36}*288b
4-fold quotients : {4,27}*216
6-fold quotients : {4,18}*144b
9-fold quotients : {4,12}*96b
12-fold quotients : {4,9}*72
18-fold quotients : {4,6}*48c
36-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,216}*1728c, {4,216}*1728d, {4,108}*1728b
Permutation Representation (GAP) :
s0 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320)
(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)(334,336)
(337,339)(338,340)(341,343)(342,344)(345,347)(346,348)(349,351)(350,352)
(353,355)(354,356)(357,359)(358,360)(361,363)(362,364)(365,367)(366,368)
(369,371)(370,372)(373,375)(374,376)(377,379)(378,380)(381,383)(382,384)
(385,387)(386,388)(389,391)(390,392)(393,395)(394,396)(397,399)(398,400)
(401,403)(402,404)(405,407)(406,408)(409,411)(410,412)(413,415)(414,416)
(417,419)(418,420)(421,423)(422,424)(425,427)(426,428)(429,431)(430,432);;
s1 := ( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 33)( 14, 35)( 15, 34)
( 16, 36)( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 25)( 22, 27)( 23, 26)
( 24, 28)( 37,105)( 38,107)( 39,106)( 40,108)( 41,101)( 42,103)( 43,102)
( 44,104)( 45, 97)( 46, 99)( 47, 98)( 48,100)( 49, 93)( 50, 95)( 51, 94)
( 52, 96)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)( 58, 87)( 59, 86)
( 60, 88)( 61, 81)( 62, 83)( 63, 82)( 64, 84)( 65, 77)( 66, 79)( 67, 78)
( 68, 80)( 69, 73)( 70, 75)( 71, 74)( 72, 76)(110,111)(113,117)(114,119)
(115,118)(116,120)(121,141)(122,143)(123,142)(124,144)(125,137)(126,139)
(127,138)(128,140)(129,133)(130,135)(131,134)(132,136)(145,213)(146,215)
(147,214)(148,216)(149,209)(150,211)(151,210)(152,212)(153,205)(154,207)
(155,206)(156,208)(157,201)(158,203)(159,202)(160,204)(161,197)(162,199)
(163,198)(164,200)(165,193)(166,195)(167,194)(168,196)(169,189)(170,191)
(171,190)(172,192)(173,185)(174,187)(175,186)(176,188)(177,181)(178,183)
(179,182)(180,184)(217,325)(218,327)(219,326)(220,328)(221,333)(222,335)
(223,334)(224,336)(225,329)(226,331)(227,330)(228,332)(229,357)(230,359)
(231,358)(232,360)(233,353)(234,355)(235,354)(236,356)(237,349)(238,351)
(239,350)(240,352)(241,345)(242,347)(243,346)(244,348)(245,341)(246,343)
(247,342)(248,344)(249,337)(250,339)(251,338)(252,340)(253,429)(254,431)
(255,430)(256,432)(257,425)(258,427)(259,426)(260,428)(261,421)(262,423)
(263,422)(264,424)(265,417)(266,419)(267,418)(268,420)(269,413)(270,415)
(271,414)(272,416)(273,409)(274,411)(275,410)(276,412)(277,405)(278,407)
(279,406)(280,408)(281,401)(282,403)(283,402)(284,404)(285,397)(286,399)
(287,398)(288,400)(289,393)(290,395)(291,394)(292,396)(293,389)(294,391)
(295,390)(296,392)(297,385)(298,387)(299,386)(300,388)(301,381)(302,383)
(303,382)(304,384)(305,377)(306,379)(307,378)(308,380)(309,373)(310,375)
(311,374)(312,376)(313,369)(314,371)(315,370)(316,372)(317,365)(318,367)
(319,366)(320,368)(321,361)(322,363)(323,362)(324,364);;
s2 := ( 1,289)( 2,292)( 3,291)( 4,290)( 5,297)( 6,300)( 7,299)( 8,298)
( 9,293)( 10,296)( 11,295)( 12,294)( 13,321)( 14,324)( 15,323)( 16,322)
( 17,317)( 18,320)( 19,319)( 20,318)( 21,313)( 22,316)( 23,315)( 24,314)
( 25,309)( 26,312)( 27,311)( 28,310)( 29,305)( 30,308)( 31,307)( 32,306)
( 33,301)( 34,304)( 35,303)( 36,302)( 37,253)( 38,256)( 39,255)( 40,254)
( 41,261)( 42,264)( 43,263)( 44,262)( 45,257)( 46,260)( 47,259)( 48,258)
( 49,285)( 50,288)( 51,287)( 52,286)( 53,281)( 54,284)( 55,283)( 56,282)
( 57,277)( 58,280)( 59,279)( 60,278)( 61,273)( 62,276)( 63,275)( 64,274)
( 65,269)( 66,272)( 67,271)( 68,270)( 69,265)( 70,268)( 71,267)( 72,266)
( 73,217)( 74,220)( 75,219)( 76,218)( 77,225)( 78,228)( 79,227)( 80,226)
( 81,221)( 82,224)( 83,223)( 84,222)( 85,249)( 86,252)( 87,251)( 88,250)
( 89,245)( 90,248)( 91,247)( 92,246)( 93,241)( 94,244)( 95,243)( 96,242)
( 97,237)( 98,240)( 99,239)(100,238)(101,233)(102,236)(103,235)(104,234)
(105,229)(106,232)(107,231)(108,230)(109,397)(110,400)(111,399)(112,398)
(113,405)(114,408)(115,407)(116,406)(117,401)(118,404)(119,403)(120,402)
(121,429)(122,432)(123,431)(124,430)(125,425)(126,428)(127,427)(128,426)
(129,421)(130,424)(131,423)(132,422)(133,417)(134,420)(135,419)(136,418)
(137,413)(138,416)(139,415)(140,414)(141,409)(142,412)(143,411)(144,410)
(145,361)(146,364)(147,363)(148,362)(149,369)(150,372)(151,371)(152,370)
(153,365)(154,368)(155,367)(156,366)(157,393)(158,396)(159,395)(160,394)
(161,389)(162,392)(163,391)(164,390)(165,385)(166,388)(167,387)(168,386)
(169,381)(170,384)(171,383)(172,382)(173,377)(174,380)(175,379)(176,378)
(177,373)(178,376)(179,375)(180,374)(181,325)(182,328)(183,327)(184,326)
(185,333)(186,336)(187,335)(188,334)(189,329)(190,332)(191,331)(192,330)
(193,357)(194,360)(195,359)(196,358)(197,353)(198,356)(199,355)(200,354)
(201,349)(202,352)(203,351)(204,350)(205,345)(206,348)(207,347)(208,346)
(209,341)(210,344)(211,343)(212,342)(213,337)(214,340)(215,339)(216,338);;
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,
s0*s1*s2*s1*s0*s1*s2*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)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320)(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)
(334,336)(337,339)(338,340)(341,343)(342,344)(345,347)(346,348)(349,351)
(350,352)(353,355)(354,356)(357,359)(358,360)(361,363)(362,364)(365,367)
(366,368)(369,371)(370,372)(373,375)(374,376)(377,379)(378,380)(381,383)
(382,384)(385,387)(386,388)(389,391)(390,392)(393,395)(394,396)(397,399)
(398,400)(401,403)(402,404)(405,407)(406,408)(409,411)(410,412)(413,415)
(414,416)(417,419)(418,420)(421,423)(422,424)(425,427)(426,428)(429,431)
(430,432);
s1 := Sym(432)!( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 33)( 14, 35)
( 15, 34)( 16, 36)( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 25)( 22, 27)
( 23, 26)( 24, 28)( 37,105)( 38,107)( 39,106)( 40,108)( 41,101)( 42,103)
( 43,102)( 44,104)( 45, 97)( 46, 99)( 47, 98)( 48,100)( 49, 93)( 50, 95)
( 51, 94)( 52, 96)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)( 58, 87)
( 59, 86)( 60, 88)( 61, 81)( 62, 83)( 63, 82)( 64, 84)( 65, 77)( 66, 79)
( 67, 78)( 68, 80)( 69, 73)( 70, 75)( 71, 74)( 72, 76)(110,111)(113,117)
(114,119)(115,118)(116,120)(121,141)(122,143)(123,142)(124,144)(125,137)
(126,139)(127,138)(128,140)(129,133)(130,135)(131,134)(132,136)(145,213)
(146,215)(147,214)(148,216)(149,209)(150,211)(151,210)(152,212)(153,205)
(154,207)(155,206)(156,208)(157,201)(158,203)(159,202)(160,204)(161,197)
(162,199)(163,198)(164,200)(165,193)(166,195)(167,194)(168,196)(169,189)
(170,191)(171,190)(172,192)(173,185)(174,187)(175,186)(176,188)(177,181)
(178,183)(179,182)(180,184)(217,325)(218,327)(219,326)(220,328)(221,333)
(222,335)(223,334)(224,336)(225,329)(226,331)(227,330)(228,332)(229,357)
(230,359)(231,358)(232,360)(233,353)(234,355)(235,354)(236,356)(237,349)
(238,351)(239,350)(240,352)(241,345)(242,347)(243,346)(244,348)(245,341)
(246,343)(247,342)(248,344)(249,337)(250,339)(251,338)(252,340)(253,429)
(254,431)(255,430)(256,432)(257,425)(258,427)(259,426)(260,428)(261,421)
(262,423)(263,422)(264,424)(265,417)(266,419)(267,418)(268,420)(269,413)
(270,415)(271,414)(272,416)(273,409)(274,411)(275,410)(276,412)(277,405)
(278,407)(279,406)(280,408)(281,401)(282,403)(283,402)(284,404)(285,397)
(286,399)(287,398)(288,400)(289,393)(290,395)(291,394)(292,396)(293,389)
(294,391)(295,390)(296,392)(297,385)(298,387)(299,386)(300,388)(301,381)
(302,383)(303,382)(304,384)(305,377)(306,379)(307,378)(308,380)(309,373)
(310,375)(311,374)(312,376)(313,369)(314,371)(315,370)(316,372)(317,365)
(318,367)(319,366)(320,368)(321,361)(322,363)(323,362)(324,364);
s2 := Sym(432)!( 1,289)( 2,292)( 3,291)( 4,290)( 5,297)( 6,300)( 7,299)
( 8,298)( 9,293)( 10,296)( 11,295)( 12,294)( 13,321)( 14,324)( 15,323)
( 16,322)( 17,317)( 18,320)( 19,319)( 20,318)( 21,313)( 22,316)( 23,315)
( 24,314)( 25,309)( 26,312)( 27,311)( 28,310)( 29,305)( 30,308)( 31,307)
( 32,306)( 33,301)( 34,304)( 35,303)( 36,302)( 37,253)( 38,256)( 39,255)
( 40,254)( 41,261)( 42,264)( 43,263)( 44,262)( 45,257)( 46,260)( 47,259)
( 48,258)( 49,285)( 50,288)( 51,287)( 52,286)( 53,281)( 54,284)( 55,283)
( 56,282)( 57,277)( 58,280)( 59,279)( 60,278)( 61,273)( 62,276)( 63,275)
( 64,274)( 65,269)( 66,272)( 67,271)( 68,270)( 69,265)( 70,268)( 71,267)
( 72,266)( 73,217)( 74,220)( 75,219)( 76,218)( 77,225)( 78,228)( 79,227)
( 80,226)( 81,221)( 82,224)( 83,223)( 84,222)( 85,249)( 86,252)( 87,251)
( 88,250)( 89,245)( 90,248)( 91,247)( 92,246)( 93,241)( 94,244)( 95,243)
( 96,242)( 97,237)( 98,240)( 99,239)(100,238)(101,233)(102,236)(103,235)
(104,234)(105,229)(106,232)(107,231)(108,230)(109,397)(110,400)(111,399)
(112,398)(113,405)(114,408)(115,407)(116,406)(117,401)(118,404)(119,403)
(120,402)(121,429)(122,432)(123,431)(124,430)(125,425)(126,428)(127,427)
(128,426)(129,421)(130,424)(131,423)(132,422)(133,417)(134,420)(135,419)
(136,418)(137,413)(138,416)(139,415)(140,414)(141,409)(142,412)(143,411)
(144,410)(145,361)(146,364)(147,363)(148,362)(149,369)(150,372)(151,371)
(152,370)(153,365)(154,368)(155,367)(156,366)(157,393)(158,396)(159,395)
(160,394)(161,389)(162,392)(163,391)(164,390)(165,385)(166,388)(167,387)
(168,386)(169,381)(170,384)(171,383)(172,382)(173,377)(174,380)(175,379)
(176,378)(177,373)(178,376)(179,375)(180,374)(181,325)(182,328)(183,327)
(184,326)(185,333)(186,336)(187,335)(188,334)(189,329)(190,332)(191,331)
(192,330)(193,357)(194,360)(195,359)(196,358)(197,353)(198,356)(199,355)
(200,354)(201,349)(202,352)(203,351)(204,350)(205,345)(206,348)(207,347)
(208,346)(209,341)(210,344)(211,343)(212,342)(213,337)(214,340)(215,339)
(216,338);
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,
s0*s1*s2*s1*s0*s1*s2*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