Polytope of Type {2,4,108}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,108}*1728c
if this polytope has a name.
Group : SmallGroup(1728,11356)
Rank : 4
Schlafli Type : {2,4,108}
Number of vertices, edges, etc : 2, 4, 216, 108
Order of s0s1s2s3 : 108
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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 : {2,4,54}*864b
3-fold quotients : {2,4,36}*576c
4-fold quotients : {2,4,27}*432
6-fold quotients : {2,4,18}*288b
9-fold quotients : {2,4,12}*192c
12-fold quotients : {2,4,9}*144
18-fold quotients : {2,4,6}*96c
36-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,113)( 4,114)( 5,111)( 6,112)( 7,117)( 8,118)( 9,115)( 10,116)
( 11,121)( 12,122)( 13,119)( 14,120)( 15,125)( 16,126)( 17,123)( 18,124)
( 19,129)( 20,130)( 21,127)( 22,128)( 23,133)( 24,134)( 25,131)( 26,132)
( 27,137)( 28,138)( 29,135)( 30,136)( 31,141)( 32,142)( 33,139)( 34,140)
( 35,145)( 36,146)( 37,143)( 38,144)( 39,149)( 40,150)( 41,147)( 42,148)
( 43,153)( 44,154)( 45,151)( 46,152)( 47,157)( 48,158)( 49,155)( 50,156)
( 51,161)( 52,162)( 53,159)( 54,160)( 55,165)( 56,166)( 57,163)( 58,164)
( 59,169)( 60,170)( 61,167)( 62,168)( 63,173)( 64,174)( 65,171)( 66,172)
( 67,177)( 68,178)( 69,175)( 70,176)( 71,181)( 72,182)( 73,179)( 74,180)
( 75,185)( 76,186)( 77,183)( 78,184)( 79,189)( 80,190)( 81,187)( 82,188)
( 83,193)( 84,194)( 85,191)( 86,192)( 87,197)( 88,198)( 89,195)( 90,196)
( 91,201)( 92,202)( 93,199)( 94,200)( 95,205)( 96,206)( 97,203)( 98,204)
( 99,209)(100,210)(101,207)(102,208)(103,213)(104,214)(105,211)(106,212)
(107,217)(108,218)(109,215)(110,216)(219,329)(220,330)(221,327)(222,328)
(223,333)(224,334)(225,331)(226,332)(227,337)(228,338)(229,335)(230,336)
(231,341)(232,342)(233,339)(234,340)(235,345)(236,346)(237,343)(238,344)
(239,349)(240,350)(241,347)(242,348)(243,353)(244,354)(245,351)(246,352)
(247,357)(248,358)(249,355)(250,356)(251,361)(252,362)(253,359)(254,360)
(255,365)(256,366)(257,363)(258,364)(259,369)(260,370)(261,367)(262,368)
(263,373)(264,374)(265,371)(266,372)(267,377)(268,378)(269,375)(270,376)
(271,381)(272,382)(273,379)(274,380)(275,385)(276,386)(277,383)(278,384)
(279,389)(280,390)(281,387)(282,388)(283,393)(284,394)(285,391)(286,392)
(287,397)(288,398)(289,395)(290,396)(291,401)(292,402)(293,399)(294,400)
(295,405)(296,406)(297,403)(298,404)(299,409)(300,410)(301,407)(302,408)
(303,413)(304,414)(305,411)(306,412)(307,417)(308,418)(309,415)(310,416)
(311,421)(312,422)(313,419)(314,420)(315,425)(316,426)(317,423)(318,424)
(319,429)(320,430)(321,427)(322,428)(323,433)(324,434)(325,431)(326,432);;
s2 := ( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 15, 35)( 16, 37)( 17, 36)
( 18, 38)( 19, 31)( 20, 33)( 21, 32)( 22, 34)( 23, 27)( 24, 29)( 25, 28)
( 26, 30)( 39,107)( 40,109)( 41,108)( 42,110)( 43,103)( 44,105)( 45,104)
( 46,106)( 47, 99)( 48,101)( 49,100)( 50,102)( 51, 95)( 52, 97)( 53, 96)
( 54, 98)( 55, 91)( 56, 93)( 57, 92)( 58, 94)( 59, 87)( 60, 89)( 61, 88)
( 62, 90)( 63, 83)( 64, 85)( 65, 84)( 66, 86)( 67, 79)( 68, 81)( 69, 80)
( 70, 82)( 71, 75)( 72, 77)( 73, 76)( 74, 78)(112,113)(115,119)(116,121)
(117,120)(118,122)(123,143)(124,145)(125,144)(126,146)(127,139)(128,141)
(129,140)(130,142)(131,135)(132,137)(133,136)(134,138)(147,215)(148,217)
(149,216)(150,218)(151,211)(152,213)(153,212)(154,214)(155,207)(156,209)
(157,208)(158,210)(159,203)(160,205)(161,204)(162,206)(163,199)(164,201)
(165,200)(166,202)(167,195)(168,197)(169,196)(170,198)(171,191)(172,193)
(173,192)(174,194)(175,187)(176,189)(177,188)(178,190)(179,183)(180,185)
(181,184)(182,186)(219,327)(220,329)(221,328)(222,330)(223,335)(224,337)
(225,336)(226,338)(227,331)(228,333)(229,332)(230,334)(231,359)(232,361)
(233,360)(234,362)(235,355)(236,357)(237,356)(238,358)(239,351)(240,353)
(241,352)(242,354)(243,347)(244,349)(245,348)(246,350)(247,343)(248,345)
(249,344)(250,346)(251,339)(252,341)(253,340)(254,342)(255,431)(256,433)
(257,432)(258,434)(259,427)(260,429)(261,428)(262,430)(263,423)(264,425)
(265,424)(266,426)(267,419)(268,421)(269,420)(270,422)(271,415)(272,417)
(273,416)(274,418)(275,411)(276,413)(277,412)(278,414)(279,407)(280,409)
(281,408)(282,410)(283,403)(284,405)(285,404)(286,406)(287,399)(288,401)
(289,400)(290,402)(291,395)(292,397)(293,396)(294,398)(295,391)(296,393)
(297,392)(298,394)(299,387)(300,389)(301,388)(302,390)(303,383)(304,385)
(305,384)(306,386)(307,379)(308,381)(309,380)(310,382)(311,375)(312,377)
(313,376)(314,378)(315,371)(316,373)(317,372)(318,374)(319,367)(320,369)
(321,368)(322,370)(323,363)(324,365)(325,364)(326,366);;
s3 := ( 3,291)( 4,294)( 5,293)( 6,292)( 7,299)( 8,302)( 9,301)( 10,300)
( 11,295)( 12,298)( 13,297)( 14,296)( 15,323)( 16,326)( 17,325)( 18,324)
( 19,319)( 20,322)( 21,321)( 22,320)( 23,315)( 24,318)( 25,317)( 26,316)
( 27,311)( 28,314)( 29,313)( 30,312)( 31,307)( 32,310)( 33,309)( 34,308)
( 35,303)( 36,306)( 37,305)( 38,304)( 39,255)( 40,258)( 41,257)( 42,256)
( 43,263)( 44,266)( 45,265)( 46,264)( 47,259)( 48,262)( 49,261)( 50,260)
( 51,287)( 52,290)( 53,289)( 54,288)( 55,283)( 56,286)( 57,285)( 58,284)
( 59,279)( 60,282)( 61,281)( 62,280)( 63,275)( 64,278)( 65,277)( 66,276)
( 67,271)( 68,274)( 69,273)( 70,272)( 71,267)( 72,270)( 73,269)( 74,268)
( 75,219)( 76,222)( 77,221)( 78,220)( 79,227)( 80,230)( 81,229)( 82,228)
( 83,223)( 84,226)( 85,225)( 86,224)( 87,251)( 88,254)( 89,253)( 90,252)
( 91,247)( 92,250)( 93,249)( 94,248)( 95,243)( 96,246)( 97,245)( 98,244)
( 99,239)(100,242)(101,241)(102,240)(103,235)(104,238)(105,237)(106,236)
(107,231)(108,234)(109,233)(110,232)(111,399)(112,402)(113,401)(114,400)
(115,407)(116,410)(117,409)(118,408)(119,403)(120,406)(121,405)(122,404)
(123,431)(124,434)(125,433)(126,432)(127,427)(128,430)(129,429)(130,428)
(131,423)(132,426)(133,425)(134,424)(135,419)(136,422)(137,421)(138,420)
(139,415)(140,418)(141,417)(142,416)(143,411)(144,414)(145,413)(146,412)
(147,363)(148,366)(149,365)(150,364)(151,371)(152,374)(153,373)(154,372)
(155,367)(156,370)(157,369)(158,368)(159,395)(160,398)(161,397)(162,396)
(163,391)(164,394)(165,393)(166,392)(167,387)(168,390)(169,389)(170,388)
(171,383)(172,386)(173,385)(174,384)(175,379)(176,382)(177,381)(178,380)
(179,375)(180,378)(181,377)(182,376)(183,327)(184,330)(185,329)(186,328)
(187,335)(188,338)(189,337)(190,336)(191,331)(192,334)(193,333)(194,332)
(195,359)(196,362)(197,361)(198,360)(199,355)(200,358)(201,357)(202,356)
(203,351)(204,354)(205,353)(206,352)(207,347)(208,350)(209,349)(210,348)
(211,343)(212,346)(213,345)(214,344)(215,339)(216,342)(217,341)(218,340);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(434)!(1,2);
s1 := Sym(434)!( 3,113)( 4,114)( 5,111)( 6,112)( 7,117)( 8,118)( 9,115)
( 10,116)( 11,121)( 12,122)( 13,119)( 14,120)( 15,125)( 16,126)( 17,123)
( 18,124)( 19,129)( 20,130)( 21,127)( 22,128)( 23,133)( 24,134)( 25,131)
( 26,132)( 27,137)( 28,138)( 29,135)( 30,136)( 31,141)( 32,142)( 33,139)
( 34,140)( 35,145)( 36,146)( 37,143)( 38,144)( 39,149)( 40,150)( 41,147)
( 42,148)( 43,153)( 44,154)( 45,151)( 46,152)( 47,157)( 48,158)( 49,155)
( 50,156)( 51,161)( 52,162)( 53,159)( 54,160)( 55,165)( 56,166)( 57,163)
( 58,164)( 59,169)( 60,170)( 61,167)( 62,168)( 63,173)( 64,174)( 65,171)
( 66,172)( 67,177)( 68,178)( 69,175)( 70,176)( 71,181)( 72,182)( 73,179)
( 74,180)( 75,185)( 76,186)( 77,183)( 78,184)( 79,189)( 80,190)( 81,187)
( 82,188)( 83,193)( 84,194)( 85,191)( 86,192)( 87,197)( 88,198)( 89,195)
( 90,196)( 91,201)( 92,202)( 93,199)( 94,200)( 95,205)( 96,206)( 97,203)
( 98,204)( 99,209)(100,210)(101,207)(102,208)(103,213)(104,214)(105,211)
(106,212)(107,217)(108,218)(109,215)(110,216)(219,329)(220,330)(221,327)
(222,328)(223,333)(224,334)(225,331)(226,332)(227,337)(228,338)(229,335)
(230,336)(231,341)(232,342)(233,339)(234,340)(235,345)(236,346)(237,343)
(238,344)(239,349)(240,350)(241,347)(242,348)(243,353)(244,354)(245,351)
(246,352)(247,357)(248,358)(249,355)(250,356)(251,361)(252,362)(253,359)
(254,360)(255,365)(256,366)(257,363)(258,364)(259,369)(260,370)(261,367)
(262,368)(263,373)(264,374)(265,371)(266,372)(267,377)(268,378)(269,375)
(270,376)(271,381)(272,382)(273,379)(274,380)(275,385)(276,386)(277,383)
(278,384)(279,389)(280,390)(281,387)(282,388)(283,393)(284,394)(285,391)
(286,392)(287,397)(288,398)(289,395)(290,396)(291,401)(292,402)(293,399)
(294,400)(295,405)(296,406)(297,403)(298,404)(299,409)(300,410)(301,407)
(302,408)(303,413)(304,414)(305,411)(306,412)(307,417)(308,418)(309,415)
(310,416)(311,421)(312,422)(313,419)(314,420)(315,425)(316,426)(317,423)
(318,424)(319,429)(320,430)(321,427)(322,428)(323,433)(324,434)(325,431)
(326,432);
s2 := Sym(434)!( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 15, 35)( 16, 37)
( 17, 36)( 18, 38)( 19, 31)( 20, 33)( 21, 32)( 22, 34)( 23, 27)( 24, 29)
( 25, 28)( 26, 30)( 39,107)( 40,109)( 41,108)( 42,110)( 43,103)( 44,105)
( 45,104)( 46,106)( 47, 99)( 48,101)( 49,100)( 50,102)( 51, 95)( 52, 97)
( 53, 96)( 54, 98)( 55, 91)( 56, 93)( 57, 92)( 58, 94)( 59, 87)( 60, 89)
( 61, 88)( 62, 90)( 63, 83)( 64, 85)( 65, 84)( 66, 86)( 67, 79)( 68, 81)
( 69, 80)( 70, 82)( 71, 75)( 72, 77)( 73, 76)( 74, 78)(112,113)(115,119)
(116,121)(117,120)(118,122)(123,143)(124,145)(125,144)(126,146)(127,139)
(128,141)(129,140)(130,142)(131,135)(132,137)(133,136)(134,138)(147,215)
(148,217)(149,216)(150,218)(151,211)(152,213)(153,212)(154,214)(155,207)
(156,209)(157,208)(158,210)(159,203)(160,205)(161,204)(162,206)(163,199)
(164,201)(165,200)(166,202)(167,195)(168,197)(169,196)(170,198)(171,191)
(172,193)(173,192)(174,194)(175,187)(176,189)(177,188)(178,190)(179,183)
(180,185)(181,184)(182,186)(219,327)(220,329)(221,328)(222,330)(223,335)
(224,337)(225,336)(226,338)(227,331)(228,333)(229,332)(230,334)(231,359)
(232,361)(233,360)(234,362)(235,355)(236,357)(237,356)(238,358)(239,351)
(240,353)(241,352)(242,354)(243,347)(244,349)(245,348)(246,350)(247,343)
(248,345)(249,344)(250,346)(251,339)(252,341)(253,340)(254,342)(255,431)
(256,433)(257,432)(258,434)(259,427)(260,429)(261,428)(262,430)(263,423)
(264,425)(265,424)(266,426)(267,419)(268,421)(269,420)(270,422)(271,415)
(272,417)(273,416)(274,418)(275,411)(276,413)(277,412)(278,414)(279,407)
(280,409)(281,408)(282,410)(283,403)(284,405)(285,404)(286,406)(287,399)
(288,401)(289,400)(290,402)(291,395)(292,397)(293,396)(294,398)(295,391)
(296,393)(297,392)(298,394)(299,387)(300,389)(301,388)(302,390)(303,383)
(304,385)(305,384)(306,386)(307,379)(308,381)(309,380)(310,382)(311,375)
(312,377)(313,376)(314,378)(315,371)(316,373)(317,372)(318,374)(319,367)
(320,369)(321,368)(322,370)(323,363)(324,365)(325,364)(326,366);
s3 := Sym(434)!( 3,291)( 4,294)( 5,293)( 6,292)( 7,299)( 8,302)( 9,301)
( 10,300)( 11,295)( 12,298)( 13,297)( 14,296)( 15,323)( 16,326)( 17,325)
( 18,324)( 19,319)( 20,322)( 21,321)( 22,320)( 23,315)( 24,318)( 25,317)
( 26,316)( 27,311)( 28,314)( 29,313)( 30,312)( 31,307)( 32,310)( 33,309)
( 34,308)( 35,303)( 36,306)( 37,305)( 38,304)( 39,255)( 40,258)( 41,257)
( 42,256)( 43,263)( 44,266)( 45,265)( 46,264)( 47,259)( 48,262)( 49,261)
( 50,260)( 51,287)( 52,290)( 53,289)( 54,288)( 55,283)( 56,286)( 57,285)
( 58,284)( 59,279)( 60,282)( 61,281)( 62,280)( 63,275)( 64,278)( 65,277)
( 66,276)( 67,271)( 68,274)( 69,273)( 70,272)( 71,267)( 72,270)( 73,269)
( 74,268)( 75,219)( 76,222)( 77,221)( 78,220)( 79,227)( 80,230)( 81,229)
( 82,228)( 83,223)( 84,226)( 85,225)( 86,224)( 87,251)( 88,254)( 89,253)
( 90,252)( 91,247)( 92,250)( 93,249)( 94,248)( 95,243)( 96,246)( 97,245)
( 98,244)( 99,239)(100,242)(101,241)(102,240)(103,235)(104,238)(105,237)
(106,236)(107,231)(108,234)(109,233)(110,232)(111,399)(112,402)(113,401)
(114,400)(115,407)(116,410)(117,409)(118,408)(119,403)(120,406)(121,405)
(122,404)(123,431)(124,434)(125,433)(126,432)(127,427)(128,430)(129,429)
(130,428)(131,423)(132,426)(133,425)(134,424)(135,419)(136,422)(137,421)
(138,420)(139,415)(140,418)(141,417)(142,416)(143,411)(144,414)(145,413)
(146,412)(147,363)(148,366)(149,365)(150,364)(151,371)(152,374)(153,373)
(154,372)(155,367)(156,370)(157,369)(158,368)(159,395)(160,398)(161,397)
(162,396)(163,391)(164,394)(165,393)(166,392)(167,387)(168,390)(169,389)
(170,388)(171,383)(172,386)(173,385)(174,384)(175,379)(176,382)(177,381)
(178,380)(179,375)(180,378)(181,377)(182,376)(183,327)(184,330)(185,329)
(186,328)(187,335)(188,338)(189,337)(190,336)(191,331)(192,334)(193,333)
(194,332)(195,359)(196,362)(197,361)(198,360)(199,355)(200,358)(201,357)
(202,356)(203,351)(204,354)(205,353)(206,352)(207,347)(208,350)(209,349)
(210,348)(211,343)(212,346)(213,345)(214,344)(215,339)(216,342)(217,341)
(218,340);
poly := sub<Sym(434)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope