Polytope of Type {6,36}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,36}*1728b
if this polytope has a name.
Group : SmallGroup(1728,30174)
Rank : 3
Schlafli Type : {6,36}
Number of vertices, edges, etc : 24, 432, 144
Order of s0s1s2 : 36
Order of s0s1s2s1 : 4
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 : {6,12}*576b
4-fold quotients : {6,36}*432a, {6,36}*432c
6-fold quotients : {6,12}*288a
8-fold quotients : {6,18}*216a
9-fold quotients : {6,4}*192b
12-fold quotients : {2,36}*144, {6,12}*144a, {6,12}*144d
18-fold quotients : {6,4}*96
24-fold quotients : {2,18}*72, {6,6}*72a
36-fold quotients : {2,12}*48, {6,4}*48a, {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
108-fold quotients : {2,4}*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 := ( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)
( 35, 36)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)
( 44, 79)( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 85)( 50, 86)( 51, 88)
( 52, 87)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)
( 60, 95)( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)( 67,104)
( 68,103)( 69,105)( 70,106)( 71,108)( 72,107)(111,112)(115,116)(119,120)
(123,124)(127,128)(131,132)(135,136)(139,140)(143,144)(145,181)(146,182)
(147,184)(148,183)(149,185)(150,186)(151,188)(152,187)(153,189)(154,190)
(155,192)(156,191)(157,193)(158,194)(159,196)(160,195)(161,197)(162,198)
(163,200)(164,199)(165,201)(166,202)(167,204)(168,203)(169,205)(170,206)
(171,208)(172,207)(173,209)(174,210)(175,212)(176,211)(177,213)(178,214)
(179,216)(180,215)(219,220)(223,224)(227,228)(231,232)(235,236)(239,240)
(243,244)(247,248)(251,252)(253,289)(254,290)(255,292)(256,291)(257,293)
(258,294)(259,296)(260,295)(261,297)(262,298)(263,300)(264,299)(265,301)
(266,302)(267,304)(268,303)(269,305)(270,306)(271,308)(272,307)(273,309)
(274,310)(275,312)(276,311)(277,313)(278,314)(279,316)(280,315)(281,317)
(282,318)(283,320)(284,319)(285,321)(286,322)(287,324)(288,323)(327,328)
(331,332)(335,336)(339,340)(343,344)(347,348)(351,352)(355,356)(359,360)
(361,397)(362,398)(363,400)(364,399)(365,401)(366,402)(367,404)(368,403)
(369,405)(370,406)(371,408)(372,407)(373,409)(374,410)(375,412)(376,411)
(377,413)(378,414)(379,416)(380,415)(381,417)(382,418)(383,420)(384,419)
(385,421)(386,422)(387,424)(388,423)(389,425)(390,426)(391,428)(392,427)
(393,429)(394,430)(395,432)(396,431);;
s1 := ( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 45)( 6, 48)( 7, 47)( 8, 46)
( 9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 69)( 14, 72)( 15, 71)( 16, 70)
( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 61)( 22, 64)( 23, 63)( 24, 62)
( 25, 57)( 26, 60)( 27, 59)( 28, 58)( 29, 53)( 30, 56)( 31, 55)( 32, 54)
( 33, 49)( 34, 52)( 35, 51)( 36, 50)( 74, 76)( 77, 81)( 78, 84)( 79, 83)
( 80, 82)( 85,105)( 86,108)( 87,107)( 88,106)( 89,101)( 90,104)( 91,103)
( 92,102)( 93, 97)( 94,100)( 95, 99)( 96, 98)(109,145)(110,148)(111,147)
(112,146)(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)(119,151)
(120,150)(121,177)(122,180)(123,179)(124,178)(125,173)(126,176)(127,175)
(128,174)(129,169)(130,172)(131,171)(132,170)(133,165)(134,168)(135,167)
(136,166)(137,161)(138,164)(139,163)(140,162)(141,157)(142,160)(143,159)
(144,158)(182,184)(185,189)(186,192)(187,191)(188,190)(193,213)(194,216)
(195,215)(196,214)(197,209)(198,212)(199,211)(200,210)(201,205)(202,208)
(203,207)(204,206)(217,361)(218,364)(219,363)(220,362)(221,369)(222,372)
(223,371)(224,370)(225,365)(226,368)(227,367)(228,366)(229,393)(230,396)
(231,395)(232,394)(233,389)(234,392)(235,391)(236,390)(237,385)(238,388)
(239,387)(240,386)(241,381)(242,384)(243,383)(244,382)(245,377)(246,380)
(247,379)(248,378)(249,373)(250,376)(251,375)(252,374)(253,325)(254,328)
(255,327)(256,326)(257,333)(258,336)(259,335)(260,334)(261,329)(262,332)
(263,331)(264,330)(265,357)(266,360)(267,359)(268,358)(269,353)(270,356)
(271,355)(272,354)(273,349)(274,352)(275,351)(276,350)(277,345)(278,348)
(279,347)(280,346)(281,341)(282,344)(283,343)(284,342)(285,337)(286,340)
(287,339)(288,338)(289,397)(290,400)(291,399)(292,398)(293,405)(294,408)
(295,407)(296,406)(297,401)(298,404)(299,403)(300,402)(301,429)(302,432)
(303,431)(304,430)(305,425)(306,428)(307,427)(308,426)(309,421)(310,424)
(311,423)(312,422)(313,417)(314,420)(315,419)(316,418)(317,413)(318,416)
(319,415)(320,414)(321,409)(322,412)(323,411)(324,410);;
s2 := ( 1,230)( 2,229)( 3,232)( 4,231)( 5,238)( 6,237)( 7,240)( 8,239)
( 9,234)( 10,233)( 11,236)( 12,235)( 13,218)( 14,217)( 15,220)( 16,219)
( 17,226)( 18,225)( 19,228)( 20,227)( 21,222)( 22,221)( 23,224)( 24,223)
( 25,250)( 26,249)( 27,252)( 28,251)( 29,246)( 30,245)( 31,248)( 32,247)
( 33,242)( 34,241)( 35,244)( 36,243)( 37,266)( 38,265)( 39,268)( 40,267)
( 41,274)( 42,273)( 43,276)( 44,275)( 45,270)( 46,269)( 47,272)( 48,271)
( 49,254)( 50,253)( 51,256)( 52,255)( 53,262)( 54,261)( 55,264)( 56,263)
( 57,258)( 58,257)( 59,260)( 60,259)( 61,286)( 62,285)( 63,288)( 64,287)
( 65,282)( 66,281)( 67,284)( 68,283)( 69,278)( 70,277)( 71,280)( 72,279)
( 73,302)( 74,301)( 75,304)( 76,303)( 77,310)( 78,309)( 79,312)( 80,311)
( 81,306)( 82,305)( 83,308)( 84,307)( 85,290)( 86,289)( 87,292)( 88,291)
( 89,298)( 90,297)( 91,300)( 92,299)( 93,294)( 94,293)( 95,296)( 96,295)
( 97,322)( 98,321)( 99,324)(100,323)(101,318)(102,317)(103,320)(104,319)
(105,314)(106,313)(107,316)(108,315)(109,338)(110,337)(111,340)(112,339)
(113,346)(114,345)(115,348)(116,347)(117,342)(118,341)(119,344)(120,343)
(121,326)(122,325)(123,328)(124,327)(125,334)(126,333)(127,336)(128,335)
(129,330)(130,329)(131,332)(132,331)(133,358)(134,357)(135,360)(136,359)
(137,354)(138,353)(139,356)(140,355)(141,350)(142,349)(143,352)(144,351)
(145,374)(146,373)(147,376)(148,375)(149,382)(150,381)(151,384)(152,383)
(153,378)(154,377)(155,380)(156,379)(157,362)(158,361)(159,364)(160,363)
(161,370)(162,369)(163,372)(164,371)(165,366)(166,365)(167,368)(168,367)
(169,394)(170,393)(171,396)(172,395)(173,390)(174,389)(175,392)(176,391)
(177,386)(178,385)(179,388)(180,387)(181,410)(182,409)(183,412)(184,411)
(185,418)(186,417)(187,420)(188,419)(189,414)(190,413)(191,416)(192,415)
(193,398)(194,397)(195,400)(196,399)(197,406)(198,405)(199,408)(200,407)
(201,402)(202,401)(203,404)(204,403)(205,430)(206,429)(207,432)(208,431)
(209,426)(210,425)(211,428)(212,427)(213,422)(214,421)(215,424)(216,423);;
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,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)
( 31, 32)( 35, 36)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)
( 43, 80)( 44, 79)( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 85)( 50, 86)
( 51, 88)( 52, 87)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 93)( 58, 94)
( 59, 96)( 60, 95)( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)
( 67,104)( 68,103)( 69,105)( 70,106)( 71,108)( 72,107)(111,112)(115,116)
(119,120)(123,124)(127,128)(131,132)(135,136)(139,140)(143,144)(145,181)
(146,182)(147,184)(148,183)(149,185)(150,186)(151,188)(152,187)(153,189)
(154,190)(155,192)(156,191)(157,193)(158,194)(159,196)(160,195)(161,197)
(162,198)(163,200)(164,199)(165,201)(166,202)(167,204)(168,203)(169,205)
(170,206)(171,208)(172,207)(173,209)(174,210)(175,212)(176,211)(177,213)
(178,214)(179,216)(180,215)(219,220)(223,224)(227,228)(231,232)(235,236)
(239,240)(243,244)(247,248)(251,252)(253,289)(254,290)(255,292)(256,291)
(257,293)(258,294)(259,296)(260,295)(261,297)(262,298)(263,300)(264,299)
(265,301)(266,302)(267,304)(268,303)(269,305)(270,306)(271,308)(272,307)
(273,309)(274,310)(275,312)(276,311)(277,313)(278,314)(279,316)(280,315)
(281,317)(282,318)(283,320)(284,319)(285,321)(286,322)(287,324)(288,323)
(327,328)(331,332)(335,336)(339,340)(343,344)(347,348)(351,352)(355,356)
(359,360)(361,397)(362,398)(363,400)(364,399)(365,401)(366,402)(367,404)
(368,403)(369,405)(370,406)(371,408)(372,407)(373,409)(374,410)(375,412)
(376,411)(377,413)(378,414)(379,416)(380,415)(381,417)(382,418)(383,420)
(384,419)(385,421)(386,422)(387,424)(388,423)(389,425)(390,426)(391,428)
(392,427)(393,429)(394,430)(395,432)(396,431);
s1 := Sym(432)!( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 45)( 6, 48)( 7, 47)
( 8, 46)( 9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 69)( 14, 72)( 15, 71)
( 16, 70)( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 61)( 22, 64)( 23, 63)
( 24, 62)( 25, 57)( 26, 60)( 27, 59)( 28, 58)( 29, 53)( 30, 56)( 31, 55)
( 32, 54)( 33, 49)( 34, 52)( 35, 51)( 36, 50)( 74, 76)( 77, 81)( 78, 84)
( 79, 83)( 80, 82)( 85,105)( 86,108)( 87,107)( 88,106)( 89,101)( 90,104)
( 91,103)( 92,102)( 93, 97)( 94,100)( 95, 99)( 96, 98)(109,145)(110,148)
(111,147)(112,146)(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)
(119,151)(120,150)(121,177)(122,180)(123,179)(124,178)(125,173)(126,176)
(127,175)(128,174)(129,169)(130,172)(131,171)(132,170)(133,165)(134,168)
(135,167)(136,166)(137,161)(138,164)(139,163)(140,162)(141,157)(142,160)
(143,159)(144,158)(182,184)(185,189)(186,192)(187,191)(188,190)(193,213)
(194,216)(195,215)(196,214)(197,209)(198,212)(199,211)(200,210)(201,205)
(202,208)(203,207)(204,206)(217,361)(218,364)(219,363)(220,362)(221,369)
(222,372)(223,371)(224,370)(225,365)(226,368)(227,367)(228,366)(229,393)
(230,396)(231,395)(232,394)(233,389)(234,392)(235,391)(236,390)(237,385)
(238,388)(239,387)(240,386)(241,381)(242,384)(243,383)(244,382)(245,377)
(246,380)(247,379)(248,378)(249,373)(250,376)(251,375)(252,374)(253,325)
(254,328)(255,327)(256,326)(257,333)(258,336)(259,335)(260,334)(261,329)
(262,332)(263,331)(264,330)(265,357)(266,360)(267,359)(268,358)(269,353)
(270,356)(271,355)(272,354)(273,349)(274,352)(275,351)(276,350)(277,345)
(278,348)(279,347)(280,346)(281,341)(282,344)(283,343)(284,342)(285,337)
(286,340)(287,339)(288,338)(289,397)(290,400)(291,399)(292,398)(293,405)
(294,408)(295,407)(296,406)(297,401)(298,404)(299,403)(300,402)(301,429)
(302,432)(303,431)(304,430)(305,425)(306,428)(307,427)(308,426)(309,421)
(310,424)(311,423)(312,422)(313,417)(314,420)(315,419)(316,418)(317,413)
(318,416)(319,415)(320,414)(321,409)(322,412)(323,411)(324,410);
s2 := Sym(432)!( 1,230)( 2,229)( 3,232)( 4,231)( 5,238)( 6,237)( 7,240)
( 8,239)( 9,234)( 10,233)( 11,236)( 12,235)( 13,218)( 14,217)( 15,220)
( 16,219)( 17,226)( 18,225)( 19,228)( 20,227)( 21,222)( 22,221)( 23,224)
( 24,223)( 25,250)( 26,249)( 27,252)( 28,251)( 29,246)( 30,245)( 31,248)
( 32,247)( 33,242)( 34,241)( 35,244)( 36,243)( 37,266)( 38,265)( 39,268)
( 40,267)( 41,274)( 42,273)( 43,276)( 44,275)( 45,270)( 46,269)( 47,272)
( 48,271)( 49,254)( 50,253)( 51,256)( 52,255)( 53,262)( 54,261)( 55,264)
( 56,263)( 57,258)( 58,257)( 59,260)( 60,259)( 61,286)( 62,285)( 63,288)
( 64,287)( 65,282)( 66,281)( 67,284)( 68,283)( 69,278)( 70,277)( 71,280)
( 72,279)( 73,302)( 74,301)( 75,304)( 76,303)( 77,310)( 78,309)( 79,312)
( 80,311)( 81,306)( 82,305)( 83,308)( 84,307)( 85,290)( 86,289)( 87,292)
( 88,291)( 89,298)( 90,297)( 91,300)( 92,299)( 93,294)( 94,293)( 95,296)
( 96,295)( 97,322)( 98,321)( 99,324)(100,323)(101,318)(102,317)(103,320)
(104,319)(105,314)(106,313)(107,316)(108,315)(109,338)(110,337)(111,340)
(112,339)(113,346)(114,345)(115,348)(116,347)(117,342)(118,341)(119,344)
(120,343)(121,326)(122,325)(123,328)(124,327)(125,334)(126,333)(127,336)
(128,335)(129,330)(130,329)(131,332)(132,331)(133,358)(134,357)(135,360)
(136,359)(137,354)(138,353)(139,356)(140,355)(141,350)(142,349)(143,352)
(144,351)(145,374)(146,373)(147,376)(148,375)(149,382)(150,381)(151,384)
(152,383)(153,378)(154,377)(155,380)(156,379)(157,362)(158,361)(159,364)
(160,363)(161,370)(162,369)(163,372)(164,371)(165,366)(166,365)(167,368)
(168,367)(169,394)(170,393)(171,396)(172,395)(173,390)(174,389)(175,392)
(176,391)(177,386)(178,385)(179,388)(180,387)(181,410)(182,409)(183,412)
(184,411)(185,418)(186,417)(187,420)(188,419)(189,414)(190,413)(191,416)
(192,415)(193,398)(194,397)(195,400)(196,399)(197,406)(198,405)(199,408)
(200,407)(201,402)(202,401)(203,404)(204,403)(205,430)(206,429)(207,432)
(208,431)(209,426)(210,425)(211,428)(212,427)(213,422)(214,421)(215,424)
(216,423);
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,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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*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 >;
References : None.
to this polytope