Polytope of Type {27,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {27,12}*1296
if this polytope has a name.
Group : SmallGroup(1296,1781)
Rank : 3
Schlafli Type : {27,12}
Number of vertices, edges, etc : 54, 324, 24
Order of s₀s₁s₂ : 54
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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) :
   3-fold quotients : {27,4}*432, {9,12}*432
   4-fold quotients : {27,6}*324
   6-fold quotients : {27,4}*216
   9-fold quotients : {9,4}*144, {3,12}*144
   12-fold quotients : {27,2}*108, {9,6}*108
   18-fold quotients : {9,4}*72
   27-fold quotients : {3,4}*48
   36-fold quotients : {9,2}*36, {3,6}*36
   54-fold quotients : {3,4}*24
   108-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 33)( 14, 34)( 15, 36)
( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)
( 24, 27)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 81)( 42, 82)( 43, 84)
( 44, 83)( 45, 77)( 46, 78)( 47, 80)( 48, 79)( 49,105)( 50,106)( 51,108)
( 52,107)( 53,101)( 54,102)( 55,104)( 56,103)( 57, 97)( 58, 98)( 59,100)
( 60, 99)( 61, 93)( 62, 94)( 63, 96)( 64, 95)( 65, 89)( 66, 90)( 67, 92)
( 68, 91)( 69, 85)( 70, 86)( 71, 88)( 72, 87)(109,249)(110,250)(111,252)
(112,251)(113,245)(114,246)(115,248)(116,247)(117,241)(118,242)(119,244)
(120,243)(121,237)(122,238)(123,240)(124,239)(125,233)(126,234)(127,236)
(128,235)(129,229)(130,230)(131,232)(132,231)(133,225)(134,226)(135,228)
(136,227)(137,221)(138,222)(139,224)(140,223)(141,217)(142,218)(143,220)
(144,219)(145,321)(146,322)(147,324)(148,323)(149,317)(150,318)(151,320)
(152,319)(153,313)(154,314)(155,316)(156,315)(157,309)(158,310)(159,312)
(160,311)(161,305)(162,306)(163,308)(164,307)(165,301)(166,302)(167,304)
(168,303)(169,297)(170,298)(171,300)(172,299)(173,293)(174,294)(175,296)
(176,295)(177,289)(178,290)(179,292)(180,291)(181,285)(182,286)(183,288)
(184,287)(185,281)(186,282)(187,284)(188,283)(189,277)(190,278)(191,280)
(192,279)(193,273)(194,274)(195,276)(196,275)(197,269)(198,270)(199,272)
(200,271)(201,265)(202,266)(203,268)(204,267)(205,261)(206,262)(207,264)
(208,263)(209,257)(210,258)(211,260)(212,259)(213,253)(214,254)(215,256)
(216,255);;
s1 := (  1,145)(  2,148)(  3,147)(  4,146)(  5,153)(  6,156)(  7,155)(  8,154)
(  9,149)( 10,152)( 11,151)( 12,150)( 13,177)( 14,180)( 15,179)( 16,178)
( 17,173)( 18,176)( 19,175)( 20,174)( 21,169)( 22,172)( 23,171)( 24,170)
( 25,165)( 26,168)( 27,167)( 28,166)( 29,161)( 30,164)( 31,163)( 32,162)
( 33,157)( 34,160)( 35,159)( 36,158)( 37,109)( 38,112)( 39,111)( 40,110)
( 41,117)( 42,120)( 43,119)( 44,118)( 45,113)( 46,116)( 47,115)( 48,114)
( 49,141)( 50,144)( 51,143)( 52,142)( 53,137)( 54,140)( 55,139)( 56,138)
( 57,133)( 58,136)( 59,135)( 60,134)( 61,129)( 62,132)( 63,131)( 64,130)
( 65,125)( 66,128)( 67,127)( 68,126)( 69,121)( 70,124)( 71,123)( 72,122)
( 73,181)( 74,184)( 75,183)( 76,182)( 77,189)( 78,192)( 79,191)( 80,190)
( 81,185)( 82,188)( 83,187)( 84,186)( 85,213)( 86,216)( 87,215)( 88,214)
( 89,209)( 90,212)( 91,211)( 92,210)( 93,205)( 94,208)( 95,207)( 96,206)
( 97,201)( 98,204)( 99,203)(100,202)(101,197)(102,200)(103,199)(104,198)
(105,193)(106,196)(107,195)(108,194)(217,285)(218,288)(219,287)(220,286)
(221,281)(222,284)(223,283)(224,282)(225,277)(226,280)(227,279)(228,278)
(229,273)(230,276)(231,275)(232,274)(233,269)(234,272)(235,271)(236,270)
(237,265)(238,268)(239,267)(240,266)(241,261)(242,264)(243,263)(244,262)
(245,257)(246,260)(247,259)(248,258)(249,253)(250,256)(251,255)(252,254)
(289,321)(290,324)(291,323)(292,322)(293,317)(294,320)(295,319)(296,318)
(297,313)(298,316)(299,315)(300,314)(301,309)(302,312)(303,311)(304,310)
(306,308);;
s2 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 74)( 38, 73)( 39, 76)( 40, 75)( 41, 78)( 42, 77)
( 43, 80)( 44, 79)( 45, 82)( 46, 81)( 47, 84)( 48, 83)( 49, 86)( 50, 85)
( 51, 88)( 52, 87)( 53, 90)( 54, 89)( 55, 92)( 56, 91)( 57, 94)( 58, 93)
( 59, 96)( 60, 95)( 61, 98)( 62, 97)( 63,100)( 64, 99)( 65,102)( 66,101)
( 67,104)( 68,103)( 69,106)( 70,105)( 71,108)( 72,107)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,182)(146,181)(147,184)(148,183)(149,186)(150,185)(151,188)(152,187)
(153,190)(154,189)(155,192)(156,191)(157,194)(158,193)(159,196)(160,195)
(161,198)(162,197)(163,200)(164,199)(165,202)(166,201)(167,204)(168,203)
(169,206)(170,205)(171,208)(172,207)(173,210)(174,209)(175,212)(176,211)
(177,214)(178,213)(179,216)(180,215)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)
(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,290)(254,289)
(255,292)(256,291)(257,294)(258,293)(259,296)(260,295)(261,298)(262,297)
(263,300)(264,299)(265,302)(266,301)(267,304)(268,303)(269,306)(270,305)
(271,308)(272,307)(273,310)(274,309)(275,312)(276,311)(277,314)(278,313)
(279,316)(280,315)(281,318)(282,317)(283,320)(284,319)(285,322)(286,321)
(287,324)(288,323);;
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*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*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*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(324)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 33)( 14, 34)
( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)
( 23, 28)( 24, 27)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 81)( 42, 82)
( 43, 84)( 44, 83)( 45, 77)( 46, 78)( 47, 80)( 48, 79)( 49,105)( 50,106)
( 51,108)( 52,107)( 53,101)( 54,102)( 55,104)( 56,103)( 57, 97)( 58, 98)
( 59,100)( 60, 99)( 61, 93)( 62, 94)( 63, 96)( 64, 95)( 65, 89)( 66, 90)
( 67, 92)( 68, 91)( 69, 85)( 70, 86)( 71, 88)( 72, 87)(109,249)(110,250)
(111,252)(112,251)(113,245)(114,246)(115,248)(116,247)(117,241)(118,242)
(119,244)(120,243)(121,237)(122,238)(123,240)(124,239)(125,233)(126,234)
(127,236)(128,235)(129,229)(130,230)(131,232)(132,231)(133,225)(134,226)
(135,228)(136,227)(137,221)(138,222)(139,224)(140,223)(141,217)(142,218)
(143,220)(144,219)(145,321)(146,322)(147,324)(148,323)(149,317)(150,318)
(151,320)(152,319)(153,313)(154,314)(155,316)(156,315)(157,309)(158,310)
(159,312)(160,311)(161,305)(162,306)(163,308)(164,307)(165,301)(166,302)
(167,304)(168,303)(169,297)(170,298)(171,300)(172,299)(173,293)(174,294)
(175,296)(176,295)(177,289)(178,290)(179,292)(180,291)(181,285)(182,286)
(183,288)(184,287)(185,281)(186,282)(187,284)(188,283)(189,277)(190,278)
(191,280)(192,279)(193,273)(194,274)(195,276)(196,275)(197,269)(198,270)
(199,272)(200,271)(201,265)(202,266)(203,268)(204,267)(205,261)(206,262)
(207,264)(208,263)(209,257)(210,258)(211,260)(212,259)(213,253)(214,254)
(215,256)(216,255);
s1 := Sym(324)!(  1,145)(  2,148)(  3,147)(  4,146)(  5,153)(  6,156)(  7,155)
(  8,154)(  9,149)( 10,152)( 11,151)( 12,150)( 13,177)( 14,180)( 15,179)
( 16,178)( 17,173)( 18,176)( 19,175)( 20,174)( 21,169)( 22,172)( 23,171)
( 24,170)( 25,165)( 26,168)( 27,167)( 28,166)( 29,161)( 30,164)( 31,163)
( 32,162)( 33,157)( 34,160)( 35,159)( 36,158)( 37,109)( 38,112)( 39,111)
( 40,110)( 41,117)( 42,120)( 43,119)( 44,118)( 45,113)( 46,116)( 47,115)
( 48,114)( 49,141)( 50,144)( 51,143)( 52,142)( 53,137)( 54,140)( 55,139)
( 56,138)( 57,133)( 58,136)( 59,135)( 60,134)( 61,129)( 62,132)( 63,131)
( 64,130)( 65,125)( 66,128)( 67,127)( 68,126)( 69,121)( 70,124)( 71,123)
( 72,122)( 73,181)( 74,184)( 75,183)( 76,182)( 77,189)( 78,192)( 79,191)
( 80,190)( 81,185)( 82,188)( 83,187)( 84,186)( 85,213)( 86,216)( 87,215)
( 88,214)( 89,209)( 90,212)( 91,211)( 92,210)( 93,205)( 94,208)( 95,207)
( 96,206)( 97,201)( 98,204)( 99,203)(100,202)(101,197)(102,200)(103,199)
(104,198)(105,193)(106,196)(107,195)(108,194)(217,285)(218,288)(219,287)
(220,286)(221,281)(222,284)(223,283)(224,282)(225,277)(226,280)(227,279)
(228,278)(229,273)(230,276)(231,275)(232,274)(233,269)(234,272)(235,271)
(236,270)(237,265)(238,268)(239,267)(240,266)(241,261)(242,264)(243,263)
(244,262)(245,257)(246,260)(247,259)(248,258)(249,253)(250,256)(251,255)
(252,254)(289,321)(290,324)(291,323)(292,322)(293,317)(294,320)(295,319)
(296,318)(297,313)(298,316)(299,315)(300,314)(301,309)(302,312)(303,311)
(304,310)(306,308);
s2 := Sym(324)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 74)( 38, 73)( 39, 76)( 40, 75)( 41, 78)
( 42, 77)( 43, 80)( 44, 79)( 45, 82)( 46, 81)( 47, 84)( 48, 83)( 49, 86)
( 50, 85)( 51, 88)( 52, 87)( 53, 90)( 54, 89)( 55, 92)( 56, 91)( 57, 94)
( 58, 93)( 59, 96)( 60, 95)( 61, 98)( 62, 97)( 63,100)( 64, 99)( 65,102)
( 66,101)( 67,104)( 68,103)( 69,106)( 70,105)( 71,108)( 72,107)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,182)(146,181)(147,184)(148,183)(149,186)(150,185)(151,188)
(152,187)(153,190)(154,189)(155,192)(156,191)(157,194)(158,193)(159,196)
(160,195)(161,198)(162,197)(163,200)(164,199)(165,202)(166,201)(167,204)
(168,203)(169,206)(170,205)(171,208)(172,207)(173,210)(174,209)(175,212)
(176,211)(177,214)(178,213)(179,216)(180,215)(217,218)(219,220)(221,222)
(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)
(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,290)
(254,289)(255,292)(256,291)(257,294)(258,293)(259,296)(260,295)(261,298)
(262,297)(263,300)(264,299)(265,302)(266,301)(267,304)(268,303)(269,306)
(270,305)(271,308)(272,307)(273,310)(274,309)(275,312)(276,311)(277,314)
(278,313)(279,316)(280,315)(281,318)(282,317)(283,320)(284,319)(285,322)
(286,321)(287,324)(288,323);
poly := sub<Sym(324)|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*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*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*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