Polytope of Type {12,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,3}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1786)
Rank : 3
Schlafli Type : {12,3}
Number of vertices, edges, etc : 216, 324, 54
Order of s₀s₁s₂ : 18
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 : {12,3}*432
   4-fold quotients : {6,3}*324
   9-fold quotients : {12,3}*144
   12-fold quotients : {6,3}*108
   27-fold quotients : {4,3}*48
   36-fold quotients : {6,3}*36
   54-fold quotients : {4,3}*24
   108-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope