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