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