Polytope of Type {72,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {72,12}*1728c
if this polytope has a name.
Group : SmallGroup(1728,5197)
Rank : 3
Schlafli Type : {72,12}
Number of vertices, edges, etc : 72, 432, 12
Order of s₀s₁s₂ : 72
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {36,12}*864a
   3-fold quotients : {72,4}*576b, {24,12}*576e
   4-fold quotients : {36,6}*432a, {18,12}*432a
   6-fold quotients : {36,4}*288a, {12,12}*288a
   8-fold quotients : {18,6}*216a
   9-fold quotients : {24,4}*192b, {8,12}*192b
   12-fold quotients : {36,2}*144, {18,4}*144a, {6,12}*144a, {12,6}*144a
   18-fold quotients : {4,12}*96a, {12,4}*96a
   24-fold quotients : {18,2}*72, {6,6}*72a
   27-fold quotients : {8,4}*64b
   36-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
   48-fold quotients : {9,2}*36
   54-fold quotients : {4,4}*32
   72-fold quotients : {2,6}*24, {6,2}*24
   108-fold quotients : {2,4}*16, {4,2}*16
   144-fold quotients : {2,3}*12, {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)( 11, 12)( 13, 18)( 14, 17)( 15, 16)
( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 29, 30)( 31, 36)( 32, 35)( 33, 34)
( 38, 39)( 40, 45)( 41, 44)( 42, 43)( 47, 48)( 49, 54)( 50, 53)( 51, 52)
( 55, 82)( 56, 84)( 57, 83)( 58, 90)( 59, 89)( 60, 88)( 61, 87)( 62, 86)
( 63, 85)( 64, 91)( 65, 93)( 66, 92)( 67, 99)( 68, 98)( 69, 97)( 70, 96)
( 71, 95)( 72, 94)( 73,100)( 74,102)( 75,101)( 76,108)( 77,107)( 78,106)
( 79,105)( 80,104)( 81,103)(109,136)(110,138)(111,137)(112,144)(113,143)
(114,142)(115,141)(116,140)(117,139)(118,145)(119,147)(120,146)(121,153)
(122,152)(123,151)(124,150)(125,149)(126,148)(127,154)(128,156)(129,155)
(130,162)(131,161)(132,160)(133,159)(134,158)(135,157)(164,165)(166,171)
(167,170)(168,169)(173,174)(175,180)(176,179)(177,178)(182,183)(184,189)
(185,188)(186,187)(191,192)(193,198)(194,197)(195,196)(200,201)(202,207)
(203,206)(204,205)(209,210)(211,216)(212,215)(213,214)(217,325)(218,327)
(219,326)(220,333)(221,332)(222,331)(223,330)(224,329)(225,328)(226,334)
(227,336)(228,335)(229,342)(230,341)(231,340)(232,339)(233,338)(234,337)
(235,343)(236,345)(237,344)(238,351)(239,350)(240,349)(241,348)(242,347)
(243,346)(244,352)(245,354)(246,353)(247,360)(248,359)(249,358)(250,357)
(251,356)(252,355)(253,361)(254,363)(255,362)(256,369)(257,368)(258,367)
(259,366)(260,365)(261,364)(262,370)(263,372)(264,371)(265,378)(266,377)
(267,376)(268,375)(269,374)(270,373)(271,406)(272,408)(273,407)(274,414)
(275,413)(276,412)(277,411)(278,410)(279,409)(280,415)(281,417)(282,416)
(283,423)(284,422)(285,421)(286,420)(287,419)(288,418)(289,424)(290,426)
(291,425)(292,432)(293,431)(294,430)(295,429)(296,428)(297,427)(298,379)
(299,381)(300,380)(301,387)(302,386)(303,385)(304,384)(305,383)(306,382)
(307,388)(308,390)(309,389)(310,396)(311,395)(312,394)(313,393)(314,392)
(315,391)(316,397)(317,399)(318,398)(319,405)(320,404)(321,403)(322,402)
(323,401)(324,400);;
s1 := (  1,220)(  2,222)(  3,221)(  4,217)(  5,219)(  6,218)(  7,225)(  8,224)
(  9,223)( 10,238)( 11,240)( 12,239)( 13,235)( 14,237)( 15,236)( 16,243)
( 17,242)( 18,241)( 19,229)( 20,231)( 21,230)( 22,226)( 23,228)( 24,227)
( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)( 32,246)
( 33,245)( 34,252)( 35,251)( 36,250)( 37,265)( 38,267)( 39,266)( 40,262)
( 41,264)( 42,263)( 43,270)( 44,269)( 45,268)( 46,256)( 47,258)( 48,257)
( 49,253)( 50,255)( 51,254)( 52,261)( 53,260)( 54,259)( 55,274)( 56,276)
( 57,275)( 58,271)( 59,273)( 60,272)( 61,279)( 62,278)( 63,277)( 64,292)
( 65,294)( 66,293)( 67,289)( 68,291)( 69,290)( 70,297)( 71,296)( 72,295)
( 73,283)( 74,285)( 75,284)( 76,280)( 77,282)( 78,281)( 79,288)( 80,287)
( 81,286)( 82,301)( 83,303)( 84,302)( 85,298)( 86,300)( 87,299)( 88,306)
( 89,305)( 90,304)( 91,319)( 92,321)( 93,320)( 94,316)( 95,318)( 96,317)
( 97,324)( 98,323)( 99,322)(100,310)(101,312)(102,311)(103,307)(104,309)
(105,308)(106,315)(107,314)(108,313)(109,355)(110,357)(111,356)(112,352)
(113,354)(114,353)(115,360)(116,359)(117,358)(118,373)(119,375)(120,374)
(121,370)(122,372)(123,371)(124,378)(125,377)(126,376)(127,364)(128,366)
(129,365)(130,361)(131,363)(132,362)(133,369)(134,368)(135,367)(136,328)
(137,330)(138,329)(139,325)(140,327)(141,326)(142,333)(143,332)(144,331)
(145,346)(146,348)(147,347)(148,343)(149,345)(150,344)(151,351)(152,350)
(153,349)(154,337)(155,339)(156,338)(157,334)(158,336)(159,335)(160,342)
(161,341)(162,340)(163,409)(164,411)(165,410)(166,406)(167,408)(168,407)
(169,414)(170,413)(171,412)(172,427)(173,429)(174,428)(175,424)(176,426)
(177,425)(178,432)(179,431)(180,430)(181,418)(182,420)(183,419)(184,415)
(185,417)(186,416)(187,423)(188,422)(189,421)(190,382)(191,384)(192,383)
(193,379)(194,381)(195,380)(196,387)(197,386)(198,385)(199,400)(200,402)
(201,401)(202,397)(203,399)(204,398)(205,405)(206,404)(207,403)(208,391)
(209,393)(210,392)(211,388)(212,390)(213,389)(214,396)(215,395)(216,394);;
s2 := (  1, 10)(  2, 11)(  3, 12)(  4, 13)(  5, 14)(  6, 15)(  7, 16)(  8, 17)
(  9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)( 34, 43)
( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)
( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)( 86, 95)
( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,145)(110,146)(111,147)(112,148)
(113,149)(114,150)(115,151)(116,152)(117,153)(118,136)(119,137)(120,138)
(121,139)(122,140)(123,141)(124,142)(125,143)(126,144)(127,154)(128,155)
(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(163,199)
(164,200)(165,201)(166,202)(167,203)(168,204)(169,205)(170,206)(171,207)
(172,190)(173,191)(174,192)(175,193)(176,194)(177,195)(178,196)(179,197)
(180,198)(181,208)(182,209)(183,210)(184,211)(185,212)(186,213)(187,214)
(188,215)(189,216)(217,280)(218,281)(219,282)(220,283)(221,284)(222,285)
(223,286)(224,287)(225,288)(226,271)(227,272)(228,273)(229,274)(230,275)
(231,276)(232,277)(233,278)(234,279)(235,289)(236,290)(237,291)(238,292)
(239,293)(240,294)(241,295)(242,296)(243,297)(244,307)(245,308)(246,309)
(247,310)(248,311)(249,312)(250,313)(251,314)(252,315)(253,298)(254,299)
(255,300)(256,301)(257,302)(258,303)(259,304)(260,305)(261,306)(262,316)
(263,317)(264,318)(265,319)(266,320)(267,321)(268,322)(269,323)(270,324)
(325,415)(326,416)(327,417)(328,418)(329,419)(330,420)(331,421)(332,422)
(333,423)(334,406)(335,407)(336,408)(337,409)(338,410)(339,411)(340,412)
(341,413)(342,414)(343,424)(344,425)(345,426)(346,427)(347,428)(348,429)
(349,430)(350,431)(351,432)(352,388)(353,389)(354,390)(355,391)(356,392)
(357,393)(358,394)(359,395)(360,396)(361,379)(362,380)(363,381)(364,382)
(365,383)(366,384)(367,385)(368,386)(369,387)(370,397)(371,398)(372,399)
(373,400)(374,401)(375,402)(376,403)(377,404)(378,405);;
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*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)( 11, 12)( 13, 18)( 14, 17)
( 15, 16)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 29, 30)( 31, 36)( 32, 35)
( 33, 34)( 38, 39)( 40, 45)( 41, 44)( 42, 43)( 47, 48)( 49, 54)( 50, 53)
( 51, 52)( 55, 82)( 56, 84)( 57, 83)( 58, 90)( 59, 89)( 60, 88)( 61, 87)
( 62, 86)( 63, 85)( 64, 91)( 65, 93)( 66, 92)( 67, 99)( 68, 98)( 69, 97)
( 70, 96)( 71, 95)( 72, 94)( 73,100)( 74,102)( 75,101)( 76,108)( 77,107)
( 78,106)( 79,105)( 80,104)( 81,103)(109,136)(110,138)(111,137)(112,144)
(113,143)(114,142)(115,141)(116,140)(117,139)(118,145)(119,147)(120,146)
(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,154)(128,156)
(129,155)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157)(164,165)
(166,171)(167,170)(168,169)(173,174)(175,180)(176,179)(177,178)(182,183)
(184,189)(185,188)(186,187)(191,192)(193,198)(194,197)(195,196)(200,201)
(202,207)(203,206)(204,205)(209,210)(211,216)(212,215)(213,214)(217,325)
(218,327)(219,326)(220,333)(221,332)(222,331)(223,330)(224,329)(225,328)
(226,334)(227,336)(228,335)(229,342)(230,341)(231,340)(232,339)(233,338)
(234,337)(235,343)(236,345)(237,344)(238,351)(239,350)(240,349)(241,348)
(242,347)(243,346)(244,352)(245,354)(246,353)(247,360)(248,359)(249,358)
(250,357)(251,356)(252,355)(253,361)(254,363)(255,362)(256,369)(257,368)
(258,367)(259,366)(260,365)(261,364)(262,370)(263,372)(264,371)(265,378)
(266,377)(267,376)(268,375)(269,374)(270,373)(271,406)(272,408)(273,407)
(274,414)(275,413)(276,412)(277,411)(278,410)(279,409)(280,415)(281,417)
(282,416)(283,423)(284,422)(285,421)(286,420)(287,419)(288,418)(289,424)
(290,426)(291,425)(292,432)(293,431)(294,430)(295,429)(296,428)(297,427)
(298,379)(299,381)(300,380)(301,387)(302,386)(303,385)(304,384)(305,383)
(306,382)(307,388)(308,390)(309,389)(310,396)(311,395)(312,394)(313,393)
(314,392)(315,391)(316,397)(317,399)(318,398)(319,405)(320,404)(321,403)
(322,402)(323,401)(324,400);
s1 := Sym(432)!(  1,220)(  2,222)(  3,221)(  4,217)(  5,219)(  6,218)(  7,225)
(  8,224)(  9,223)( 10,238)( 11,240)( 12,239)( 13,235)( 14,237)( 15,236)
( 16,243)( 17,242)( 18,241)( 19,229)( 20,231)( 21,230)( 22,226)( 23,228)
( 24,227)( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)
( 32,246)( 33,245)( 34,252)( 35,251)( 36,250)( 37,265)( 38,267)( 39,266)
( 40,262)( 41,264)( 42,263)( 43,270)( 44,269)( 45,268)( 46,256)( 47,258)
( 48,257)( 49,253)( 50,255)( 51,254)( 52,261)( 53,260)( 54,259)( 55,274)
( 56,276)( 57,275)( 58,271)( 59,273)( 60,272)( 61,279)( 62,278)( 63,277)
( 64,292)( 65,294)( 66,293)( 67,289)( 68,291)( 69,290)( 70,297)( 71,296)
( 72,295)( 73,283)( 74,285)( 75,284)( 76,280)( 77,282)( 78,281)( 79,288)
( 80,287)( 81,286)( 82,301)( 83,303)( 84,302)( 85,298)( 86,300)( 87,299)
( 88,306)( 89,305)( 90,304)( 91,319)( 92,321)( 93,320)( 94,316)( 95,318)
( 96,317)( 97,324)( 98,323)( 99,322)(100,310)(101,312)(102,311)(103,307)
(104,309)(105,308)(106,315)(107,314)(108,313)(109,355)(110,357)(111,356)
(112,352)(113,354)(114,353)(115,360)(116,359)(117,358)(118,373)(119,375)
(120,374)(121,370)(122,372)(123,371)(124,378)(125,377)(126,376)(127,364)
(128,366)(129,365)(130,361)(131,363)(132,362)(133,369)(134,368)(135,367)
(136,328)(137,330)(138,329)(139,325)(140,327)(141,326)(142,333)(143,332)
(144,331)(145,346)(146,348)(147,347)(148,343)(149,345)(150,344)(151,351)
(152,350)(153,349)(154,337)(155,339)(156,338)(157,334)(158,336)(159,335)
(160,342)(161,341)(162,340)(163,409)(164,411)(165,410)(166,406)(167,408)
(168,407)(169,414)(170,413)(171,412)(172,427)(173,429)(174,428)(175,424)
(176,426)(177,425)(178,432)(179,431)(180,430)(181,418)(182,420)(183,419)
(184,415)(185,417)(186,416)(187,423)(188,422)(189,421)(190,382)(191,384)
(192,383)(193,379)(194,381)(195,380)(196,387)(197,386)(198,385)(199,400)
(200,402)(201,401)(202,397)(203,399)(204,398)(205,405)(206,404)(207,403)
(208,391)(209,393)(210,392)(211,388)(212,390)(213,389)(214,396)(215,395)
(216,394);
s2 := Sym(432)!(  1, 10)(  2, 11)(  3, 12)(  4, 13)(  5, 14)(  6, 15)(  7, 16)
(  8, 17)(  9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)
( 34, 43)( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)
( 60, 69)( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)
( 86, 95)( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,145)(110,146)(111,147)
(112,148)(113,149)(114,150)(115,151)(116,152)(117,153)(118,136)(119,137)
(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144)(127,154)
(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)
(163,199)(164,200)(165,201)(166,202)(167,203)(168,204)(169,205)(170,206)
(171,207)(172,190)(173,191)(174,192)(175,193)(176,194)(177,195)(178,196)
(179,197)(180,198)(181,208)(182,209)(183,210)(184,211)(185,212)(186,213)
(187,214)(188,215)(189,216)(217,280)(218,281)(219,282)(220,283)(221,284)
(222,285)(223,286)(224,287)(225,288)(226,271)(227,272)(228,273)(229,274)
(230,275)(231,276)(232,277)(233,278)(234,279)(235,289)(236,290)(237,291)
(238,292)(239,293)(240,294)(241,295)(242,296)(243,297)(244,307)(245,308)
(246,309)(247,310)(248,311)(249,312)(250,313)(251,314)(252,315)(253,298)
(254,299)(255,300)(256,301)(257,302)(258,303)(259,304)(260,305)(261,306)
(262,316)(263,317)(264,318)(265,319)(266,320)(267,321)(268,322)(269,323)
(270,324)(325,415)(326,416)(327,417)(328,418)(329,419)(330,420)(331,421)
(332,422)(333,423)(334,406)(335,407)(336,408)(337,409)(338,410)(339,411)
(340,412)(341,413)(342,414)(343,424)(344,425)(345,426)(346,427)(347,428)
(348,429)(349,430)(350,431)(351,432)(352,388)(353,389)(354,390)(355,391)
(356,392)(357,393)(358,394)(359,395)(360,396)(361,379)(362,380)(363,381)
(364,382)(365,383)(366,384)(367,385)(368,386)(369,387)(370,397)(371,398)
(372,399)(373,400)(374,401)(375,402)(376,403)(377,404)(378,405);
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*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