Polytope of Type {6,72}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,72}*1728b
if this polytope has a name.
Group : SmallGroup(1728,30189)
Rank : 3
Schlafli Type : {6,72}
Number of vertices, edges, etc : 12, 432, 144
Order of s0s1s2 : 18
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,36}*864
3-fold quotients : {6,24}*576c
4-fold quotients : {6,36}*432c
6-fold quotients : {6,12}*288a
8-fold quotients : {6,18}*216a
9-fold quotients : {6,8}*192c
12-fold quotients : {6,12}*144d
18-fold quotients : {6,4}*96
24-fold quotients : {2,18}*72, {6,6}*72a
36-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
48-fold quotients : {2,9}*36
72-fold quotients : {3,4}*24, {2,6}*24, {6,2}*24
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 := ( 3, 4)( 5, 7)( 6, 8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)( 21, 23)
( 22, 24)( 27, 28)( 29, 31)( 30, 32)( 35, 36)( 37, 39)( 38, 40)( 43, 44)
( 45, 47)( 46, 48)( 51, 52)( 53, 55)( 54, 56)( 59, 60)( 61, 63)( 62, 64)
( 67, 68)( 69, 71)( 70, 72)( 73,145)( 74,146)( 75,148)( 76,147)( 77,151)
( 78,152)( 79,149)( 80,150)( 81,153)( 82,154)( 83,156)( 84,155)( 85,159)
( 86,160)( 87,157)( 88,158)( 89,161)( 90,162)( 91,164)( 92,163)( 93,167)
( 94,168)( 95,165)( 96,166)( 97,169)( 98,170)( 99,172)(100,171)(101,175)
(102,176)(103,173)(104,174)(105,177)(106,178)(107,180)(108,179)(109,183)
(110,184)(111,181)(112,182)(113,185)(114,186)(115,188)(116,187)(117,191)
(118,192)(119,189)(120,190)(121,193)(122,194)(123,196)(124,195)(125,199)
(126,200)(127,197)(128,198)(129,201)(130,202)(131,204)(132,203)(133,207)
(134,208)(135,205)(136,206)(137,209)(138,210)(139,212)(140,211)(141,215)
(142,216)(143,213)(144,214)(217,218)(221,224)(222,223)(225,226)(229,232)
(230,231)(233,234)(237,240)(238,239)(241,242)(245,248)(246,247)(249,250)
(253,256)(254,255)(257,258)(261,264)(262,263)(265,266)(269,272)(270,271)
(273,274)(277,280)(278,279)(281,282)(285,288)(286,287)(289,362)(290,361)
(291,363)(292,364)(293,368)(294,367)(295,366)(296,365)(297,370)(298,369)
(299,371)(300,372)(301,376)(302,375)(303,374)(304,373)(305,378)(306,377)
(307,379)(308,380)(309,384)(310,383)(311,382)(312,381)(313,386)(314,385)
(315,387)(316,388)(317,392)(318,391)(319,390)(320,389)(321,394)(322,393)
(323,395)(324,396)(325,400)(326,399)(327,398)(328,397)(329,402)(330,401)
(331,403)(332,404)(333,408)(334,407)(335,406)(336,405)(337,410)(338,409)
(339,411)(340,412)(341,416)(342,415)(343,414)(344,413)(345,418)(346,417)
(347,419)(348,420)(349,424)(350,423)(351,422)(352,421)(353,426)(354,425)
(355,427)(356,428)(357,432)(358,431)(359,430)(360,429);;
s1 := ( 1, 73)( 2, 74)( 3, 79)( 4, 80)( 5, 78)( 6, 77)( 7, 75)( 8, 76)
( 9, 89)( 10, 90)( 11, 95)( 12, 96)( 13, 94)( 14, 93)( 15, 91)( 16, 92)
( 17, 81)( 18, 82)( 19, 87)( 20, 88)( 21, 86)( 22, 85)( 23, 83)( 24, 84)
( 25,137)( 26,138)( 27,143)( 28,144)( 29,142)( 30,141)( 31,139)( 32,140)
( 33,129)( 34,130)( 35,135)( 36,136)( 37,134)( 38,133)( 39,131)( 40,132)
( 41,121)( 42,122)( 43,127)( 44,128)( 45,126)( 46,125)( 47,123)( 48,124)
( 49,113)( 50,114)( 51,119)( 52,120)( 53,118)( 54,117)( 55,115)( 56,116)
( 57,105)( 58,106)( 59,111)( 60,112)( 61,110)( 62,109)( 63,107)( 64,108)
( 65, 97)( 66, 98)( 67,103)( 68,104)( 69,102)( 70,101)( 71, 99)( 72,100)
(147,151)(148,152)(149,150)(153,161)(154,162)(155,167)(156,168)(157,166)
(158,165)(159,163)(160,164)(169,209)(170,210)(171,215)(172,216)(173,214)
(174,213)(175,211)(176,212)(177,201)(178,202)(179,207)(180,208)(181,206)
(182,205)(183,203)(184,204)(185,193)(186,194)(187,199)(188,200)(189,198)
(190,197)(191,195)(192,196)(217,289)(218,290)(219,295)(220,296)(221,294)
(222,293)(223,291)(224,292)(225,305)(226,306)(227,311)(228,312)(229,310)
(230,309)(231,307)(232,308)(233,297)(234,298)(235,303)(236,304)(237,302)
(238,301)(239,299)(240,300)(241,353)(242,354)(243,359)(244,360)(245,358)
(246,357)(247,355)(248,356)(249,345)(250,346)(251,351)(252,352)(253,350)
(254,349)(255,347)(256,348)(257,337)(258,338)(259,343)(260,344)(261,342)
(262,341)(263,339)(264,340)(265,329)(266,330)(267,335)(268,336)(269,334)
(270,333)(271,331)(272,332)(273,321)(274,322)(275,327)(276,328)(277,326)
(278,325)(279,323)(280,324)(281,313)(282,314)(283,319)(284,320)(285,318)
(286,317)(287,315)(288,316)(363,367)(364,368)(365,366)(369,377)(370,378)
(371,383)(372,384)(373,382)(374,381)(375,379)(376,380)(385,425)(386,426)
(387,431)(388,432)(389,430)(390,429)(391,427)(392,428)(393,417)(394,418)
(395,423)(396,424)(397,422)(398,421)(399,419)(400,420)(401,409)(402,410)
(403,415)(404,416)(405,414)(406,413)(407,411)(408,412);;
s2 := ( 1,243)( 2,244)( 3,242)( 4,241)( 5,247)( 6,248)( 7,246)( 8,245)
( 9,259)( 10,260)( 11,258)( 12,257)( 13,263)( 14,264)( 15,262)( 16,261)
( 17,251)( 18,252)( 19,250)( 20,249)( 21,255)( 22,256)( 23,254)( 24,253)
( 25,219)( 26,220)( 27,218)( 28,217)( 29,223)( 30,224)( 31,222)( 32,221)
( 33,235)( 34,236)( 35,234)( 36,233)( 37,239)( 38,240)( 39,238)( 40,237)
( 41,227)( 42,228)( 43,226)( 44,225)( 45,231)( 46,232)( 47,230)( 48,229)
( 49,283)( 50,284)( 51,282)( 52,281)( 53,287)( 54,288)( 55,286)( 56,285)
( 57,275)( 58,276)( 59,274)( 60,273)( 61,279)( 62,280)( 63,278)( 64,277)
( 65,267)( 66,268)( 67,266)( 68,265)( 69,271)( 70,272)( 71,270)( 72,269)
( 73,315)( 74,316)( 75,314)( 76,313)( 77,319)( 78,320)( 79,318)( 80,317)
( 81,331)( 82,332)( 83,330)( 84,329)( 85,335)( 86,336)( 87,334)( 88,333)
( 89,323)( 90,324)( 91,322)( 92,321)( 93,327)( 94,328)( 95,326)( 96,325)
( 97,291)( 98,292)( 99,290)(100,289)(101,295)(102,296)(103,294)(104,293)
(105,307)(106,308)(107,306)(108,305)(109,311)(110,312)(111,310)(112,309)
(113,299)(114,300)(115,298)(116,297)(117,303)(118,304)(119,302)(120,301)
(121,355)(122,356)(123,354)(124,353)(125,359)(126,360)(127,358)(128,357)
(129,347)(130,348)(131,346)(132,345)(133,351)(134,352)(135,350)(136,349)
(137,339)(138,340)(139,338)(140,337)(141,343)(142,344)(143,342)(144,341)
(145,387)(146,388)(147,386)(148,385)(149,391)(150,392)(151,390)(152,389)
(153,403)(154,404)(155,402)(156,401)(157,407)(158,408)(159,406)(160,405)
(161,395)(162,396)(163,394)(164,393)(165,399)(166,400)(167,398)(168,397)
(169,363)(170,364)(171,362)(172,361)(173,367)(174,368)(175,366)(176,365)
(177,379)(178,380)(179,378)(180,377)(181,383)(182,384)(183,382)(184,381)
(185,371)(186,372)(187,370)(188,369)(189,375)(190,376)(191,374)(192,373)
(193,427)(194,428)(195,426)(196,425)(197,431)(198,432)(199,430)(200,429)
(201,419)(202,420)(203,418)(204,417)(205,423)(206,424)(207,422)(208,421)
(209,411)(210,412)(211,410)(212,409)(213,415)(214,416)(215,414)(216,413);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 3, 4)( 5, 7)( 6, 8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)
( 21, 23)( 22, 24)( 27, 28)( 29, 31)( 30, 32)( 35, 36)( 37, 39)( 38, 40)
( 43, 44)( 45, 47)( 46, 48)( 51, 52)( 53, 55)( 54, 56)( 59, 60)( 61, 63)
( 62, 64)( 67, 68)( 69, 71)( 70, 72)( 73,145)( 74,146)( 75,148)( 76,147)
( 77,151)( 78,152)( 79,149)( 80,150)( 81,153)( 82,154)( 83,156)( 84,155)
( 85,159)( 86,160)( 87,157)( 88,158)( 89,161)( 90,162)( 91,164)( 92,163)
( 93,167)( 94,168)( 95,165)( 96,166)( 97,169)( 98,170)( 99,172)(100,171)
(101,175)(102,176)(103,173)(104,174)(105,177)(106,178)(107,180)(108,179)
(109,183)(110,184)(111,181)(112,182)(113,185)(114,186)(115,188)(116,187)
(117,191)(118,192)(119,189)(120,190)(121,193)(122,194)(123,196)(124,195)
(125,199)(126,200)(127,197)(128,198)(129,201)(130,202)(131,204)(132,203)
(133,207)(134,208)(135,205)(136,206)(137,209)(138,210)(139,212)(140,211)
(141,215)(142,216)(143,213)(144,214)(217,218)(221,224)(222,223)(225,226)
(229,232)(230,231)(233,234)(237,240)(238,239)(241,242)(245,248)(246,247)
(249,250)(253,256)(254,255)(257,258)(261,264)(262,263)(265,266)(269,272)
(270,271)(273,274)(277,280)(278,279)(281,282)(285,288)(286,287)(289,362)
(290,361)(291,363)(292,364)(293,368)(294,367)(295,366)(296,365)(297,370)
(298,369)(299,371)(300,372)(301,376)(302,375)(303,374)(304,373)(305,378)
(306,377)(307,379)(308,380)(309,384)(310,383)(311,382)(312,381)(313,386)
(314,385)(315,387)(316,388)(317,392)(318,391)(319,390)(320,389)(321,394)
(322,393)(323,395)(324,396)(325,400)(326,399)(327,398)(328,397)(329,402)
(330,401)(331,403)(332,404)(333,408)(334,407)(335,406)(336,405)(337,410)
(338,409)(339,411)(340,412)(341,416)(342,415)(343,414)(344,413)(345,418)
(346,417)(347,419)(348,420)(349,424)(350,423)(351,422)(352,421)(353,426)
(354,425)(355,427)(356,428)(357,432)(358,431)(359,430)(360,429);
s1 := Sym(432)!( 1, 73)( 2, 74)( 3, 79)( 4, 80)( 5, 78)( 6, 77)( 7, 75)
( 8, 76)( 9, 89)( 10, 90)( 11, 95)( 12, 96)( 13, 94)( 14, 93)( 15, 91)
( 16, 92)( 17, 81)( 18, 82)( 19, 87)( 20, 88)( 21, 86)( 22, 85)( 23, 83)
( 24, 84)( 25,137)( 26,138)( 27,143)( 28,144)( 29,142)( 30,141)( 31,139)
( 32,140)( 33,129)( 34,130)( 35,135)( 36,136)( 37,134)( 38,133)( 39,131)
( 40,132)( 41,121)( 42,122)( 43,127)( 44,128)( 45,126)( 46,125)( 47,123)
( 48,124)( 49,113)( 50,114)( 51,119)( 52,120)( 53,118)( 54,117)( 55,115)
( 56,116)( 57,105)( 58,106)( 59,111)( 60,112)( 61,110)( 62,109)( 63,107)
( 64,108)( 65, 97)( 66, 98)( 67,103)( 68,104)( 69,102)( 70,101)( 71, 99)
( 72,100)(147,151)(148,152)(149,150)(153,161)(154,162)(155,167)(156,168)
(157,166)(158,165)(159,163)(160,164)(169,209)(170,210)(171,215)(172,216)
(173,214)(174,213)(175,211)(176,212)(177,201)(178,202)(179,207)(180,208)
(181,206)(182,205)(183,203)(184,204)(185,193)(186,194)(187,199)(188,200)
(189,198)(190,197)(191,195)(192,196)(217,289)(218,290)(219,295)(220,296)
(221,294)(222,293)(223,291)(224,292)(225,305)(226,306)(227,311)(228,312)
(229,310)(230,309)(231,307)(232,308)(233,297)(234,298)(235,303)(236,304)
(237,302)(238,301)(239,299)(240,300)(241,353)(242,354)(243,359)(244,360)
(245,358)(246,357)(247,355)(248,356)(249,345)(250,346)(251,351)(252,352)
(253,350)(254,349)(255,347)(256,348)(257,337)(258,338)(259,343)(260,344)
(261,342)(262,341)(263,339)(264,340)(265,329)(266,330)(267,335)(268,336)
(269,334)(270,333)(271,331)(272,332)(273,321)(274,322)(275,327)(276,328)
(277,326)(278,325)(279,323)(280,324)(281,313)(282,314)(283,319)(284,320)
(285,318)(286,317)(287,315)(288,316)(363,367)(364,368)(365,366)(369,377)
(370,378)(371,383)(372,384)(373,382)(374,381)(375,379)(376,380)(385,425)
(386,426)(387,431)(388,432)(389,430)(390,429)(391,427)(392,428)(393,417)
(394,418)(395,423)(396,424)(397,422)(398,421)(399,419)(400,420)(401,409)
(402,410)(403,415)(404,416)(405,414)(406,413)(407,411)(408,412);
s2 := Sym(432)!( 1,243)( 2,244)( 3,242)( 4,241)( 5,247)( 6,248)( 7,246)
( 8,245)( 9,259)( 10,260)( 11,258)( 12,257)( 13,263)( 14,264)( 15,262)
( 16,261)( 17,251)( 18,252)( 19,250)( 20,249)( 21,255)( 22,256)( 23,254)
( 24,253)( 25,219)( 26,220)( 27,218)( 28,217)( 29,223)( 30,224)( 31,222)
( 32,221)( 33,235)( 34,236)( 35,234)( 36,233)( 37,239)( 38,240)( 39,238)
( 40,237)( 41,227)( 42,228)( 43,226)( 44,225)( 45,231)( 46,232)( 47,230)
( 48,229)( 49,283)( 50,284)( 51,282)( 52,281)( 53,287)( 54,288)( 55,286)
( 56,285)( 57,275)( 58,276)( 59,274)( 60,273)( 61,279)( 62,280)( 63,278)
( 64,277)( 65,267)( 66,268)( 67,266)( 68,265)( 69,271)( 70,272)( 71,270)
( 72,269)( 73,315)( 74,316)( 75,314)( 76,313)( 77,319)( 78,320)( 79,318)
( 80,317)( 81,331)( 82,332)( 83,330)( 84,329)( 85,335)( 86,336)( 87,334)
( 88,333)( 89,323)( 90,324)( 91,322)( 92,321)( 93,327)( 94,328)( 95,326)
( 96,325)( 97,291)( 98,292)( 99,290)(100,289)(101,295)(102,296)(103,294)
(104,293)(105,307)(106,308)(107,306)(108,305)(109,311)(110,312)(111,310)
(112,309)(113,299)(114,300)(115,298)(116,297)(117,303)(118,304)(119,302)
(120,301)(121,355)(122,356)(123,354)(124,353)(125,359)(126,360)(127,358)
(128,357)(129,347)(130,348)(131,346)(132,345)(133,351)(134,352)(135,350)
(136,349)(137,339)(138,340)(139,338)(140,337)(141,343)(142,344)(143,342)
(144,341)(145,387)(146,388)(147,386)(148,385)(149,391)(150,392)(151,390)
(152,389)(153,403)(154,404)(155,402)(156,401)(157,407)(158,408)(159,406)
(160,405)(161,395)(162,396)(163,394)(164,393)(165,399)(166,400)(167,398)
(168,397)(169,363)(170,364)(171,362)(172,361)(173,367)(174,368)(175,366)
(176,365)(177,379)(178,380)(179,378)(180,377)(181,383)(182,384)(183,382)
(184,381)(185,371)(186,372)(187,370)(188,369)(189,375)(190,376)(191,374)
(192,373)(193,427)(194,428)(195,426)(196,425)(197,431)(198,432)(199,430)
(200,429)(201,419)(202,420)(203,418)(204,417)(205,423)(206,424)(207,422)
(208,421)(209,411)(210,412)(211,410)(212,409)(213,415)(214,416)(215,414)
(216,413);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2 >;
References : None.
to this polytope