Polytope of Type {4,36,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,36,6}*1728e
if this polytope has a name.
Group : SmallGroup(1728,30228)
Rank : 4
Schlafli Type : {4,36,6}
Number of vertices, edges, etc : 4, 72, 108, 6
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
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 : {4,18,6}*864c
3-fold quotients : {4,36,2}*576c, {4,12,6}*576f
6-fold quotients : {4,18,2}*288b, {4,6,6}*288d
9-fold quotients : {4,12,2}*192c
12-fold quotients : {4,9,2}*144
18-fold quotients : {4,6,2}*96c
36-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,111)( 2,112)( 3,109)( 4,110)( 5,115)( 6,116)( 7,113)( 8,114)
( 9,119)( 10,120)( 11,117)( 12,118)( 13,123)( 14,124)( 15,121)( 16,122)
( 17,127)( 18,128)( 19,125)( 20,126)( 21,131)( 22,132)( 23,129)( 24,130)
( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)( 32,138)
( 33,143)( 34,144)( 35,141)( 36,142)( 37,147)( 38,148)( 39,145)( 40,146)
( 41,151)( 42,152)( 43,149)( 44,150)( 45,155)( 46,156)( 47,153)( 48,154)
( 49,159)( 50,160)( 51,157)( 52,158)( 53,163)( 54,164)( 55,161)( 56,162)
( 57,167)( 58,168)( 59,165)( 60,166)( 61,171)( 62,172)( 63,169)( 64,170)
( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)( 72,178)
( 73,183)( 74,184)( 75,181)( 76,182)( 77,187)( 78,188)( 79,185)( 80,186)
( 81,191)( 82,192)( 83,189)( 84,190)( 85,195)( 86,196)( 87,193)( 88,194)
( 89,199)( 90,200)( 91,197)( 92,198)( 93,203)( 94,204)( 95,201)( 96,202)
( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)(104,210)
(105,215)(106,216)(107,213)(108,214)(217,327)(218,328)(219,325)(220,326)
(221,331)(222,332)(223,329)(224,330)(225,335)(226,336)(227,333)(228,334)
(229,339)(230,340)(231,337)(232,338)(233,343)(234,344)(235,341)(236,342)
(237,347)(238,348)(239,345)(240,346)(241,351)(242,352)(243,349)(244,350)
(245,355)(246,356)(247,353)(248,354)(249,359)(250,360)(251,357)(252,358)
(253,363)(254,364)(255,361)(256,362)(257,367)(258,368)(259,365)(260,366)
(261,371)(262,372)(263,369)(264,370)(265,375)(266,376)(267,373)(268,374)
(269,379)(270,380)(271,377)(272,378)(273,383)(274,384)(275,381)(276,382)
(277,387)(278,388)(279,385)(280,386)(281,391)(282,392)(283,389)(284,390)
(285,395)(286,396)(287,393)(288,394)(289,399)(290,400)(291,397)(292,398)
(293,403)(294,404)(295,401)(296,402)(297,407)(298,408)(299,405)(300,406)
(301,411)(302,412)(303,409)(304,410)(305,415)(306,416)(307,413)(308,414)
(309,419)(310,420)(311,417)(312,418)(313,423)(314,424)(315,421)(316,422)
(317,427)(318,428)(319,425)(320,426)(321,431)(322,432)(323,429)(324,430);;
s1 := ( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 37, 81)
( 38, 82)( 39, 84)( 40, 83)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 73)
( 46, 74)( 47, 76)( 48, 75)( 49, 93)( 50, 94)( 51, 96)( 52, 95)( 53, 89)
( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)( 60, 87)( 61,105)
( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)( 67,104)( 68,103)( 69, 97)
( 70, 98)( 71,100)( 72, 99)(111,112)(113,117)(114,118)(115,120)(116,119)
(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)(137,141)(138,142)
(139,144)(140,143)(145,189)(146,190)(147,192)(148,191)(149,185)(150,186)
(151,188)(152,187)(153,181)(154,182)(155,184)(156,183)(157,201)(158,202)
(159,204)(160,203)(161,197)(162,198)(163,200)(164,199)(165,193)(166,194)
(167,196)(168,195)(169,213)(170,214)(171,216)(172,215)(173,209)(174,210)
(175,212)(176,211)(177,205)(178,206)(179,208)(180,207)(217,325)(218,326)
(219,328)(220,327)(221,333)(222,334)(223,336)(224,335)(225,329)(226,330)
(227,332)(228,331)(229,337)(230,338)(231,340)(232,339)(233,345)(234,346)
(235,348)(236,347)(237,341)(238,342)(239,344)(240,343)(241,349)(242,350)
(243,352)(244,351)(245,357)(246,358)(247,360)(248,359)(249,353)(250,354)
(251,356)(252,355)(253,405)(254,406)(255,408)(256,407)(257,401)(258,402)
(259,404)(260,403)(261,397)(262,398)(263,400)(264,399)(265,417)(266,418)
(267,420)(268,419)(269,413)(270,414)(271,416)(272,415)(273,409)(274,410)
(275,412)(276,411)(277,429)(278,430)(279,432)(280,431)(281,425)(282,426)
(283,428)(284,427)(285,421)(286,422)(287,424)(288,423)(289,369)(290,370)
(291,372)(292,371)(293,365)(294,366)(295,368)(296,367)(297,361)(298,362)
(299,364)(300,363)(301,381)(302,382)(303,384)(304,383)(305,377)(306,378)
(307,380)(308,379)(309,373)(310,374)(311,376)(312,375)(313,393)(314,394)
(315,396)(316,395)(317,389)(318,390)(319,392)(320,391)(321,385)(322,386)
(323,388)(324,387);;
s2 := ( 1,253)( 2,256)( 3,255)( 4,254)( 5,261)( 6,264)( 7,263)( 8,262)
( 9,257)( 10,260)( 11,259)( 12,258)( 13,277)( 14,280)( 15,279)( 16,278)
( 17,285)( 18,288)( 19,287)( 20,286)( 21,281)( 22,284)( 23,283)( 24,282)
( 25,265)( 26,268)( 27,267)( 28,266)( 29,273)( 30,276)( 31,275)( 32,274)
( 33,269)( 34,272)( 35,271)( 36,270)( 37,217)( 38,220)( 39,219)( 40,218)
( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)( 48,222)
( 49,241)( 50,244)( 51,243)( 52,242)( 53,249)( 54,252)( 55,251)( 56,250)
( 57,245)( 58,248)( 59,247)( 60,246)( 61,229)( 62,232)( 63,231)( 64,230)
( 65,237)( 66,240)( 67,239)( 68,238)( 69,233)( 70,236)( 71,235)( 72,234)
( 73,297)( 74,300)( 75,299)( 76,298)( 77,293)( 78,296)( 79,295)( 80,294)
( 81,289)( 82,292)( 83,291)( 84,290)( 85,321)( 86,324)( 87,323)( 88,322)
( 89,317)( 90,320)( 91,319)( 92,318)( 93,313)( 94,316)( 95,315)( 96,314)
( 97,309)( 98,312)( 99,311)(100,310)(101,305)(102,308)(103,307)(104,306)
(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)(112,362)
(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)(120,366)
(121,385)(122,388)(123,387)(124,386)(125,393)(126,396)(127,395)(128,394)
(129,389)(130,392)(131,391)(132,390)(133,373)(134,376)(135,375)(136,374)
(137,381)(138,384)(139,383)(140,382)(141,377)(142,380)(143,379)(144,378)
(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)(152,334)
(153,329)(154,332)(155,331)(156,330)(157,349)(158,352)(159,351)(160,350)
(161,357)(162,360)(163,359)(164,358)(165,353)(166,356)(167,355)(168,354)
(169,337)(170,340)(171,339)(172,338)(173,345)(174,348)(175,347)(176,346)
(177,341)(178,344)(179,343)(180,342)(181,405)(182,408)(183,407)(184,406)
(185,401)(186,404)(187,403)(188,402)(189,397)(190,400)(191,399)(192,398)
(193,429)(194,432)(195,431)(196,430)(197,425)(198,428)(199,427)(200,426)
(201,421)(202,424)(203,423)(204,422)(205,417)(206,420)(207,419)(208,418)
(209,413)(210,416)(211,415)(212,414)(213,409)(214,412)(215,411)(216,410);;
s3 := ( 1, 13)( 2, 14)( 3, 15)( 4, 16)( 5, 17)( 6, 18)( 7, 19)( 8, 20)
( 9, 21)( 10, 22)( 11, 23)( 12, 24)( 37, 49)( 38, 50)( 39, 51)( 40, 52)
( 41, 53)( 42, 54)( 43, 55)( 44, 56)( 45, 57)( 46, 58)( 47, 59)( 48, 60)
( 73, 85)( 74, 86)( 75, 87)( 76, 88)( 77, 89)( 78, 90)( 79, 91)( 80, 92)
( 81, 93)( 82, 94)( 83, 95)( 84, 96)(109,121)(110,122)(111,123)(112,124)
(113,125)(114,126)(115,127)(116,128)(117,129)(118,130)(119,131)(120,132)
(145,157)(146,158)(147,159)(148,160)(149,161)(150,162)(151,163)(152,164)
(153,165)(154,166)(155,167)(156,168)(181,193)(182,194)(183,195)(184,196)
(185,197)(186,198)(187,199)(188,200)(189,201)(190,202)(191,203)(192,204)
(217,229)(218,230)(219,231)(220,232)(221,233)(222,234)(223,235)(224,236)
(225,237)(226,238)(227,239)(228,240)(253,265)(254,266)(255,267)(256,268)
(257,269)(258,270)(259,271)(260,272)(261,273)(262,274)(263,275)(264,276)
(289,301)(290,302)(291,303)(292,304)(293,305)(294,306)(295,307)(296,308)
(297,309)(298,310)(299,311)(300,312)(325,337)(326,338)(327,339)(328,340)
(329,341)(330,342)(331,343)(332,344)(333,345)(334,346)(335,347)(336,348)
(361,373)(362,374)(363,375)(364,376)(365,377)(366,378)(367,379)(368,380)
(369,381)(370,382)(371,383)(372,384)(397,409)(398,410)(399,411)(400,412)
(401,413)(402,414)(403,415)(404,416)(405,417)(406,418)(407,419)(408,420);;
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*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 1,111)( 2,112)( 3,109)( 4,110)( 5,115)( 6,116)( 7,113)
( 8,114)( 9,119)( 10,120)( 11,117)( 12,118)( 13,123)( 14,124)( 15,121)
( 16,122)( 17,127)( 18,128)( 19,125)( 20,126)( 21,131)( 22,132)( 23,129)
( 24,130)( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)
( 32,138)( 33,143)( 34,144)( 35,141)( 36,142)( 37,147)( 38,148)( 39,145)
( 40,146)( 41,151)( 42,152)( 43,149)( 44,150)( 45,155)( 46,156)( 47,153)
( 48,154)( 49,159)( 50,160)( 51,157)( 52,158)( 53,163)( 54,164)( 55,161)
( 56,162)( 57,167)( 58,168)( 59,165)( 60,166)( 61,171)( 62,172)( 63,169)
( 64,170)( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)
( 72,178)( 73,183)( 74,184)( 75,181)( 76,182)( 77,187)( 78,188)( 79,185)
( 80,186)( 81,191)( 82,192)( 83,189)( 84,190)( 85,195)( 86,196)( 87,193)
( 88,194)( 89,199)( 90,200)( 91,197)( 92,198)( 93,203)( 94,204)( 95,201)
( 96,202)( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)
(104,210)(105,215)(106,216)(107,213)(108,214)(217,327)(218,328)(219,325)
(220,326)(221,331)(222,332)(223,329)(224,330)(225,335)(226,336)(227,333)
(228,334)(229,339)(230,340)(231,337)(232,338)(233,343)(234,344)(235,341)
(236,342)(237,347)(238,348)(239,345)(240,346)(241,351)(242,352)(243,349)
(244,350)(245,355)(246,356)(247,353)(248,354)(249,359)(250,360)(251,357)
(252,358)(253,363)(254,364)(255,361)(256,362)(257,367)(258,368)(259,365)
(260,366)(261,371)(262,372)(263,369)(264,370)(265,375)(266,376)(267,373)
(268,374)(269,379)(270,380)(271,377)(272,378)(273,383)(274,384)(275,381)
(276,382)(277,387)(278,388)(279,385)(280,386)(281,391)(282,392)(283,389)
(284,390)(285,395)(286,396)(287,393)(288,394)(289,399)(290,400)(291,397)
(292,398)(293,403)(294,404)(295,401)(296,402)(297,407)(298,408)(299,405)
(300,406)(301,411)(302,412)(303,409)(304,410)(305,415)(306,416)(307,413)
(308,414)(309,419)(310,420)(311,417)(312,418)(313,423)(314,424)(315,421)
(316,422)(317,427)(318,428)(319,425)(320,426)(321,431)(322,432)(323,429)
(324,430);
s1 := Sym(432)!( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)
( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)
( 37, 81)( 38, 82)( 39, 84)( 40, 83)( 41, 77)( 42, 78)( 43, 80)( 44, 79)
( 45, 73)( 46, 74)( 47, 76)( 48, 75)( 49, 93)( 50, 94)( 51, 96)( 52, 95)
( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)( 60, 87)
( 61,105)( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)( 67,104)( 68,103)
( 69, 97)( 70, 98)( 71,100)( 72, 99)(111,112)(113,117)(114,118)(115,120)
(116,119)(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)(137,141)
(138,142)(139,144)(140,143)(145,189)(146,190)(147,192)(148,191)(149,185)
(150,186)(151,188)(152,187)(153,181)(154,182)(155,184)(156,183)(157,201)
(158,202)(159,204)(160,203)(161,197)(162,198)(163,200)(164,199)(165,193)
(166,194)(167,196)(168,195)(169,213)(170,214)(171,216)(172,215)(173,209)
(174,210)(175,212)(176,211)(177,205)(178,206)(179,208)(180,207)(217,325)
(218,326)(219,328)(220,327)(221,333)(222,334)(223,336)(224,335)(225,329)
(226,330)(227,332)(228,331)(229,337)(230,338)(231,340)(232,339)(233,345)
(234,346)(235,348)(236,347)(237,341)(238,342)(239,344)(240,343)(241,349)
(242,350)(243,352)(244,351)(245,357)(246,358)(247,360)(248,359)(249,353)
(250,354)(251,356)(252,355)(253,405)(254,406)(255,408)(256,407)(257,401)
(258,402)(259,404)(260,403)(261,397)(262,398)(263,400)(264,399)(265,417)
(266,418)(267,420)(268,419)(269,413)(270,414)(271,416)(272,415)(273,409)
(274,410)(275,412)(276,411)(277,429)(278,430)(279,432)(280,431)(281,425)
(282,426)(283,428)(284,427)(285,421)(286,422)(287,424)(288,423)(289,369)
(290,370)(291,372)(292,371)(293,365)(294,366)(295,368)(296,367)(297,361)
(298,362)(299,364)(300,363)(301,381)(302,382)(303,384)(304,383)(305,377)
(306,378)(307,380)(308,379)(309,373)(310,374)(311,376)(312,375)(313,393)
(314,394)(315,396)(316,395)(317,389)(318,390)(319,392)(320,391)(321,385)
(322,386)(323,388)(324,387);
s2 := Sym(432)!( 1,253)( 2,256)( 3,255)( 4,254)( 5,261)( 6,264)( 7,263)
( 8,262)( 9,257)( 10,260)( 11,259)( 12,258)( 13,277)( 14,280)( 15,279)
( 16,278)( 17,285)( 18,288)( 19,287)( 20,286)( 21,281)( 22,284)( 23,283)
( 24,282)( 25,265)( 26,268)( 27,267)( 28,266)( 29,273)( 30,276)( 31,275)
( 32,274)( 33,269)( 34,272)( 35,271)( 36,270)( 37,217)( 38,220)( 39,219)
( 40,218)( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)
( 48,222)( 49,241)( 50,244)( 51,243)( 52,242)( 53,249)( 54,252)( 55,251)
( 56,250)( 57,245)( 58,248)( 59,247)( 60,246)( 61,229)( 62,232)( 63,231)
( 64,230)( 65,237)( 66,240)( 67,239)( 68,238)( 69,233)( 70,236)( 71,235)
( 72,234)( 73,297)( 74,300)( 75,299)( 76,298)( 77,293)( 78,296)( 79,295)
( 80,294)( 81,289)( 82,292)( 83,291)( 84,290)( 85,321)( 86,324)( 87,323)
( 88,322)( 89,317)( 90,320)( 91,319)( 92,318)( 93,313)( 94,316)( 95,315)
( 96,314)( 97,309)( 98,312)( 99,311)(100,310)(101,305)(102,308)(103,307)
(104,306)(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)
(112,362)(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)
(120,366)(121,385)(122,388)(123,387)(124,386)(125,393)(126,396)(127,395)
(128,394)(129,389)(130,392)(131,391)(132,390)(133,373)(134,376)(135,375)
(136,374)(137,381)(138,384)(139,383)(140,382)(141,377)(142,380)(143,379)
(144,378)(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)
(152,334)(153,329)(154,332)(155,331)(156,330)(157,349)(158,352)(159,351)
(160,350)(161,357)(162,360)(163,359)(164,358)(165,353)(166,356)(167,355)
(168,354)(169,337)(170,340)(171,339)(172,338)(173,345)(174,348)(175,347)
(176,346)(177,341)(178,344)(179,343)(180,342)(181,405)(182,408)(183,407)
(184,406)(185,401)(186,404)(187,403)(188,402)(189,397)(190,400)(191,399)
(192,398)(193,429)(194,432)(195,431)(196,430)(197,425)(198,428)(199,427)
(200,426)(201,421)(202,424)(203,423)(204,422)(205,417)(206,420)(207,419)
(208,418)(209,413)(210,416)(211,415)(212,414)(213,409)(214,412)(215,411)
(216,410);
s3 := Sym(432)!( 1, 13)( 2, 14)( 3, 15)( 4, 16)( 5, 17)( 6, 18)( 7, 19)
( 8, 20)( 9, 21)( 10, 22)( 11, 23)( 12, 24)( 37, 49)( 38, 50)( 39, 51)
( 40, 52)( 41, 53)( 42, 54)( 43, 55)( 44, 56)( 45, 57)( 46, 58)( 47, 59)
( 48, 60)( 73, 85)( 74, 86)( 75, 87)( 76, 88)( 77, 89)( 78, 90)( 79, 91)
( 80, 92)( 81, 93)( 82, 94)( 83, 95)( 84, 96)(109,121)(110,122)(111,123)
(112,124)(113,125)(114,126)(115,127)(116,128)(117,129)(118,130)(119,131)
(120,132)(145,157)(146,158)(147,159)(148,160)(149,161)(150,162)(151,163)
(152,164)(153,165)(154,166)(155,167)(156,168)(181,193)(182,194)(183,195)
(184,196)(185,197)(186,198)(187,199)(188,200)(189,201)(190,202)(191,203)
(192,204)(217,229)(218,230)(219,231)(220,232)(221,233)(222,234)(223,235)
(224,236)(225,237)(226,238)(227,239)(228,240)(253,265)(254,266)(255,267)
(256,268)(257,269)(258,270)(259,271)(260,272)(261,273)(262,274)(263,275)
(264,276)(289,301)(290,302)(291,303)(292,304)(293,305)(294,306)(295,307)
(296,308)(297,309)(298,310)(299,311)(300,312)(325,337)(326,338)(327,339)
(328,340)(329,341)(330,342)(331,343)(332,344)(333,345)(334,346)(335,347)
(336,348)(361,373)(362,374)(363,375)(364,376)(365,377)(366,378)(367,379)
(368,380)(369,381)(370,382)(371,383)(372,384)(397,409)(398,410)(399,411)
(400,412)(401,413)(402,414)(403,415)(404,416)(405,417)(406,418)(407,419)
(408,420);
poly := sub<Sym(432)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2 >;
References : None.
to this polytope