Polytope of Type {8,27}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,27}*1728
if this polytope has a name.
Group : SmallGroup(1728,2198)
Rank : 3
Schlafli Type : {8,27}
Number of vertices, edges, etc : 32, 432, 108
Order of s0s1s2 : 54
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) :
3-fold quotients : {8,9}*576
4-fold quotients : {4,27}*432
8-fold quotients : {4,27}*216
9-fold quotients : {8,3}*192
12-fold quotients : {4,9}*144
16-fold quotients : {2,27}*108
24-fold quotients : {4,9}*72
36-fold quotients : {4,3}*48
48-fold quotients : {2,9}*36
72-fold quotients : {4,3}*24
144-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 14)( 6, 13)( 7, 16)( 8, 15)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 30)( 22, 29)( 23, 32)( 24, 31)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 46)( 38, 45)( 39, 48)( 40, 47)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 62)( 54, 61)( 55, 64)( 56, 63)
( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 78)( 70, 77)( 71, 80)( 72, 79)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 94)( 86, 93)( 87, 96)( 88, 95)
( 97,105)( 98,106)( 99,107)(100,108)(101,110)(102,109)(103,112)(104,111)
(113,121)(114,122)(115,123)(116,124)(117,126)(118,125)(119,128)(120,127)
(129,137)(130,138)(131,139)(132,140)(133,142)(134,141)(135,144)(136,143)
(145,153)(146,154)(147,155)(148,156)(149,158)(150,157)(151,160)(152,159)
(161,169)(162,170)(163,171)(164,172)(165,174)(166,173)(167,176)(168,175)
(177,185)(178,186)(179,187)(180,188)(181,190)(182,189)(183,192)(184,191)
(193,201)(194,202)(195,203)(196,204)(197,206)(198,205)(199,208)(200,207)
(209,217)(210,218)(211,219)(212,220)(213,222)(214,221)(215,224)(216,223)
(225,233)(226,234)(227,235)(228,236)(229,238)(230,237)(231,240)(232,239)
(241,249)(242,250)(243,251)(244,252)(245,254)(246,253)(247,256)(248,255)
(257,265)(258,266)(259,267)(260,268)(261,270)(262,269)(263,272)(264,271)
(273,281)(274,282)(275,283)(276,284)(277,286)(278,285)(279,288)(280,287)
(289,297)(290,298)(291,299)(292,300)(293,302)(294,301)(295,304)(296,303)
(305,313)(306,314)(307,315)(308,316)(309,318)(310,317)(311,320)(312,319)
(321,329)(322,330)(323,331)(324,332)(325,334)(326,333)(327,336)(328,335)
(337,345)(338,346)(339,347)(340,348)(341,350)(342,349)(343,352)(344,351)
(353,361)(354,362)(355,363)(356,364)(357,366)(358,365)(359,368)(360,367)
(369,377)(370,378)(371,379)(372,380)(373,382)(374,381)(375,384)(376,383)
(385,393)(386,394)(387,395)(388,396)(389,398)(390,397)(391,400)(392,399)
(401,409)(402,410)(403,411)(404,412)(405,414)(406,413)(407,416)(408,415)
(417,425)(418,426)(419,427)(420,428)(421,430)(422,429)(423,432)(424,431);;
s1 := ( 3, 4)( 5, 6)( 9, 14)( 10, 13)( 11, 15)( 12, 16)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 46)( 26, 45)
( 27, 47)( 28, 48)( 29, 42)( 30, 41)( 31, 43)( 32, 44)( 49,129)( 50,130)
( 51,132)( 52,131)( 53,134)( 54,133)( 55,135)( 56,136)( 57,142)( 58,141)
( 59,143)( 60,144)( 61,138)( 62,137)( 63,139)( 64,140)( 65,113)( 66,114)
( 67,116)( 68,115)( 69,118)( 70,117)( 71,119)( 72,120)( 73,126)( 74,125)
( 75,127)( 76,128)( 77,122)( 78,121)( 79,123)( 80,124)( 81, 97)( 82, 98)
( 83,100)( 84, 99)( 85,102)( 86,101)( 87,103)( 88,104)( 89,110)( 90,109)
( 91,111)( 92,112)( 93,106)( 94,105)( 95,107)( 96,108)(145,417)(146,418)
(147,420)(148,419)(149,422)(150,421)(151,423)(152,424)(153,430)(154,429)
(155,431)(156,432)(157,426)(158,425)(159,427)(160,428)(161,401)(162,402)
(163,404)(164,403)(165,406)(166,405)(167,407)(168,408)(169,414)(170,413)
(171,415)(172,416)(173,410)(174,409)(175,411)(176,412)(177,385)(178,386)
(179,388)(180,387)(181,390)(182,389)(183,391)(184,392)(185,398)(186,397)
(187,399)(188,400)(189,394)(190,393)(191,395)(192,396)(193,369)(194,370)
(195,372)(196,371)(197,374)(198,373)(199,375)(200,376)(201,382)(202,381)
(203,383)(204,384)(205,378)(206,377)(207,379)(208,380)(209,353)(210,354)
(211,356)(212,355)(213,358)(214,357)(215,359)(216,360)(217,366)(218,365)
(219,367)(220,368)(221,362)(222,361)(223,363)(224,364)(225,337)(226,338)
(227,340)(228,339)(229,342)(230,341)(231,343)(232,344)(233,350)(234,349)
(235,351)(236,352)(237,346)(238,345)(239,347)(240,348)(241,321)(242,322)
(243,324)(244,323)(245,326)(246,325)(247,327)(248,328)(249,334)(250,333)
(251,335)(252,336)(253,330)(254,329)(255,331)(256,332)(257,305)(258,306)
(259,308)(260,307)(261,310)(262,309)(263,311)(264,312)(265,318)(266,317)
(267,319)(268,320)(269,314)(270,313)(271,315)(272,316)(273,289)(274,290)
(275,292)(276,291)(277,294)(278,293)(279,295)(280,296)(281,302)(282,301)
(283,303)(284,304)(285,298)(286,297)(287,299)(288,300);;
s2 := ( 1,145)( 2,148)( 3,147)( 4,146)( 5,160)( 6,157)( 7,158)( 8,159)
( 9,153)( 10,156)( 11,155)( 12,154)( 13,150)( 14,151)( 15,152)( 16,149)
( 17,177)( 18,180)( 19,179)( 20,178)( 21,192)( 22,189)( 23,190)( 24,191)
( 25,185)( 26,188)( 27,187)( 28,186)( 29,182)( 30,183)( 31,184)( 32,181)
( 33,161)( 34,164)( 35,163)( 36,162)( 37,176)( 38,173)( 39,174)( 40,175)
( 41,169)( 42,172)( 43,171)( 44,170)( 45,166)( 46,167)( 47,168)( 48,165)
( 49,273)( 50,276)( 51,275)( 52,274)( 53,288)( 54,285)( 55,286)( 56,287)
( 57,281)( 58,284)( 59,283)( 60,282)( 61,278)( 62,279)( 63,280)( 64,277)
( 65,257)( 66,260)( 67,259)( 68,258)( 69,272)( 70,269)( 71,270)( 72,271)
( 73,265)( 74,268)( 75,267)( 76,266)( 77,262)( 78,263)( 79,264)( 80,261)
( 81,241)( 82,244)( 83,243)( 84,242)( 85,256)( 86,253)( 87,254)( 88,255)
( 89,249)( 90,252)( 91,251)( 92,250)( 93,246)( 94,247)( 95,248)( 96,245)
( 97,225)( 98,228)( 99,227)(100,226)(101,240)(102,237)(103,238)(104,239)
(105,233)(106,236)(107,235)(108,234)(109,230)(110,231)(111,232)(112,229)
(113,209)(114,212)(115,211)(116,210)(117,224)(118,221)(119,222)(120,223)
(121,217)(122,220)(123,219)(124,218)(125,214)(126,215)(127,216)(128,213)
(129,193)(130,196)(131,195)(132,194)(133,208)(134,205)(135,206)(136,207)
(137,201)(138,204)(139,203)(140,202)(141,198)(142,199)(143,200)(144,197)
(289,417)(290,420)(291,419)(292,418)(293,432)(294,429)(295,430)(296,431)
(297,425)(298,428)(299,427)(300,426)(301,422)(302,423)(303,424)(304,421)
(305,401)(306,404)(307,403)(308,402)(309,416)(310,413)(311,414)(312,415)
(313,409)(314,412)(315,411)(316,410)(317,406)(318,407)(319,408)(320,405)
(321,385)(322,388)(323,387)(324,386)(325,400)(326,397)(327,398)(328,399)
(329,393)(330,396)(331,395)(332,394)(333,390)(334,391)(335,392)(336,389)
(337,369)(338,372)(339,371)(340,370)(341,384)(342,381)(343,382)(344,383)
(345,377)(346,380)(347,379)(348,378)(349,374)(350,375)(351,376)(352,373)
(354,356)(357,368)(358,365)(359,366)(360,367)(362,364);;
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*s2*s1*s2*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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 14)( 6, 13)( 7, 16)
( 8, 15)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 30)( 22, 29)( 23, 32)
( 24, 31)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 46)( 38, 45)( 39, 48)
( 40, 47)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 62)( 54, 61)( 55, 64)
( 56, 63)( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 78)( 70, 77)( 71, 80)
( 72, 79)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 94)( 86, 93)( 87, 96)
( 88, 95)( 97,105)( 98,106)( 99,107)(100,108)(101,110)(102,109)(103,112)
(104,111)(113,121)(114,122)(115,123)(116,124)(117,126)(118,125)(119,128)
(120,127)(129,137)(130,138)(131,139)(132,140)(133,142)(134,141)(135,144)
(136,143)(145,153)(146,154)(147,155)(148,156)(149,158)(150,157)(151,160)
(152,159)(161,169)(162,170)(163,171)(164,172)(165,174)(166,173)(167,176)
(168,175)(177,185)(178,186)(179,187)(180,188)(181,190)(182,189)(183,192)
(184,191)(193,201)(194,202)(195,203)(196,204)(197,206)(198,205)(199,208)
(200,207)(209,217)(210,218)(211,219)(212,220)(213,222)(214,221)(215,224)
(216,223)(225,233)(226,234)(227,235)(228,236)(229,238)(230,237)(231,240)
(232,239)(241,249)(242,250)(243,251)(244,252)(245,254)(246,253)(247,256)
(248,255)(257,265)(258,266)(259,267)(260,268)(261,270)(262,269)(263,272)
(264,271)(273,281)(274,282)(275,283)(276,284)(277,286)(278,285)(279,288)
(280,287)(289,297)(290,298)(291,299)(292,300)(293,302)(294,301)(295,304)
(296,303)(305,313)(306,314)(307,315)(308,316)(309,318)(310,317)(311,320)
(312,319)(321,329)(322,330)(323,331)(324,332)(325,334)(326,333)(327,336)
(328,335)(337,345)(338,346)(339,347)(340,348)(341,350)(342,349)(343,352)
(344,351)(353,361)(354,362)(355,363)(356,364)(357,366)(358,365)(359,368)
(360,367)(369,377)(370,378)(371,379)(372,380)(373,382)(374,381)(375,384)
(376,383)(385,393)(386,394)(387,395)(388,396)(389,398)(390,397)(391,400)
(392,399)(401,409)(402,410)(403,411)(404,412)(405,414)(406,413)(407,416)
(408,415)(417,425)(418,426)(419,427)(420,428)(421,430)(422,429)(423,432)
(424,431);
s1 := Sym(432)!( 3, 4)( 5, 6)( 9, 14)( 10, 13)( 11, 15)( 12, 16)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 46)
( 26, 45)( 27, 47)( 28, 48)( 29, 42)( 30, 41)( 31, 43)( 32, 44)( 49,129)
( 50,130)( 51,132)( 52,131)( 53,134)( 54,133)( 55,135)( 56,136)( 57,142)
( 58,141)( 59,143)( 60,144)( 61,138)( 62,137)( 63,139)( 64,140)( 65,113)
( 66,114)( 67,116)( 68,115)( 69,118)( 70,117)( 71,119)( 72,120)( 73,126)
( 74,125)( 75,127)( 76,128)( 77,122)( 78,121)( 79,123)( 80,124)( 81, 97)
( 82, 98)( 83,100)( 84, 99)( 85,102)( 86,101)( 87,103)( 88,104)( 89,110)
( 90,109)( 91,111)( 92,112)( 93,106)( 94,105)( 95,107)( 96,108)(145,417)
(146,418)(147,420)(148,419)(149,422)(150,421)(151,423)(152,424)(153,430)
(154,429)(155,431)(156,432)(157,426)(158,425)(159,427)(160,428)(161,401)
(162,402)(163,404)(164,403)(165,406)(166,405)(167,407)(168,408)(169,414)
(170,413)(171,415)(172,416)(173,410)(174,409)(175,411)(176,412)(177,385)
(178,386)(179,388)(180,387)(181,390)(182,389)(183,391)(184,392)(185,398)
(186,397)(187,399)(188,400)(189,394)(190,393)(191,395)(192,396)(193,369)
(194,370)(195,372)(196,371)(197,374)(198,373)(199,375)(200,376)(201,382)
(202,381)(203,383)(204,384)(205,378)(206,377)(207,379)(208,380)(209,353)
(210,354)(211,356)(212,355)(213,358)(214,357)(215,359)(216,360)(217,366)
(218,365)(219,367)(220,368)(221,362)(222,361)(223,363)(224,364)(225,337)
(226,338)(227,340)(228,339)(229,342)(230,341)(231,343)(232,344)(233,350)
(234,349)(235,351)(236,352)(237,346)(238,345)(239,347)(240,348)(241,321)
(242,322)(243,324)(244,323)(245,326)(246,325)(247,327)(248,328)(249,334)
(250,333)(251,335)(252,336)(253,330)(254,329)(255,331)(256,332)(257,305)
(258,306)(259,308)(260,307)(261,310)(262,309)(263,311)(264,312)(265,318)
(266,317)(267,319)(268,320)(269,314)(270,313)(271,315)(272,316)(273,289)
(274,290)(275,292)(276,291)(277,294)(278,293)(279,295)(280,296)(281,302)
(282,301)(283,303)(284,304)(285,298)(286,297)(287,299)(288,300);
s2 := Sym(432)!( 1,145)( 2,148)( 3,147)( 4,146)( 5,160)( 6,157)( 7,158)
( 8,159)( 9,153)( 10,156)( 11,155)( 12,154)( 13,150)( 14,151)( 15,152)
( 16,149)( 17,177)( 18,180)( 19,179)( 20,178)( 21,192)( 22,189)( 23,190)
( 24,191)( 25,185)( 26,188)( 27,187)( 28,186)( 29,182)( 30,183)( 31,184)
( 32,181)( 33,161)( 34,164)( 35,163)( 36,162)( 37,176)( 38,173)( 39,174)
( 40,175)( 41,169)( 42,172)( 43,171)( 44,170)( 45,166)( 46,167)( 47,168)
( 48,165)( 49,273)( 50,276)( 51,275)( 52,274)( 53,288)( 54,285)( 55,286)
( 56,287)( 57,281)( 58,284)( 59,283)( 60,282)( 61,278)( 62,279)( 63,280)
( 64,277)( 65,257)( 66,260)( 67,259)( 68,258)( 69,272)( 70,269)( 71,270)
( 72,271)( 73,265)( 74,268)( 75,267)( 76,266)( 77,262)( 78,263)( 79,264)
( 80,261)( 81,241)( 82,244)( 83,243)( 84,242)( 85,256)( 86,253)( 87,254)
( 88,255)( 89,249)( 90,252)( 91,251)( 92,250)( 93,246)( 94,247)( 95,248)
( 96,245)( 97,225)( 98,228)( 99,227)(100,226)(101,240)(102,237)(103,238)
(104,239)(105,233)(106,236)(107,235)(108,234)(109,230)(110,231)(111,232)
(112,229)(113,209)(114,212)(115,211)(116,210)(117,224)(118,221)(119,222)
(120,223)(121,217)(122,220)(123,219)(124,218)(125,214)(126,215)(127,216)
(128,213)(129,193)(130,196)(131,195)(132,194)(133,208)(134,205)(135,206)
(136,207)(137,201)(138,204)(139,203)(140,202)(141,198)(142,199)(143,200)
(144,197)(289,417)(290,420)(291,419)(292,418)(293,432)(294,429)(295,430)
(296,431)(297,425)(298,428)(299,427)(300,426)(301,422)(302,423)(303,424)
(304,421)(305,401)(306,404)(307,403)(308,402)(309,416)(310,413)(311,414)
(312,415)(313,409)(314,412)(315,411)(316,410)(317,406)(318,407)(319,408)
(320,405)(321,385)(322,388)(323,387)(324,386)(325,400)(326,397)(327,398)
(328,399)(329,393)(330,396)(331,395)(332,394)(333,390)(334,391)(335,392)
(336,389)(337,369)(338,372)(339,371)(340,370)(341,384)(342,381)(343,382)
(344,383)(345,377)(346,380)(347,379)(348,378)(349,374)(350,375)(351,376)
(352,373)(354,356)(357,368)(358,365)(359,366)(360,367)(362,364);
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*s2*s1*s2*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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*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 >;
References : None.
to this polytope