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