Polytope of Type {12,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,9}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1787)
Rank : 3
Schlafli Type : {12,9}
Number of vertices, edges, etc : 72, 324, 54
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 12
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 : {12,3}*432
   4-fold quotients : {6,9}*324d
   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)(109,257)(110,258)(111,260)(112,259)(113,253)
(114,254)(115,256)(116,255)(117,261)(118,262)(119,264)(120,263)(121,269)
(122,270)(123,272)(124,271)(125,265)(126,266)(127,268)(128,267)(129,273)
(130,274)(131,276)(132,275)(133,281)(134,282)(135,284)(136,283)(137,277)
(138,278)(139,280)(140,279)(141,285)(142,286)(143,288)(144,287)(145,221)
(146,222)(147,224)(148,223)(149,217)(150,218)(151,220)(152,219)(153,225)
(154,226)(155,228)(156,227)(157,233)(158,234)(159,236)(160,235)(161,229)
(162,230)(163,232)(164,231)(165,237)(166,238)(167,240)(168,239)(169,245)
(170,246)(171,248)(172,247)(173,241)(174,242)(175,244)(176,243)(177,249)
(178,250)(179,252)(180,251)(181,293)(182,294)(183,296)(184,295)(185,289)
(186,290)(187,292)(188,291)(189,297)(190,298)(191,300)(192,299)(193,305)
(194,306)(195,308)(196,307)(197,301)(198,302)(199,304)(200,303)(201,309)
(202,310)(203,312)(204,311)(205,317)(206,318)(207,320)(208,319)(209,313)
(210,314)(211,316)(212,315)(213,321)(214,322)(215,324)(216,323);;
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)(217,221)(218,224)(219,223)(220,222)
(226,228)(230,232)(233,237)(234,240)(235,239)(236,238)(241,249)(242,252)
(243,251)(244,250)(246,248)(253,321)(254,324)(255,323)(256,322)(257,317)
(258,320)(259,319)(260,318)(261,313)(262,316)(263,315)(264,314)(265,293)
(266,296)(267,295)(268,294)(269,289)(270,292)(271,291)(272,290)(273,297)
(274,300)(275,299)(276,298)(277,301)(278,304)(279,303)(280,302)(281,309)
(282,312)(283,311)(284,310)(285,305)(286,308)(287,307)(288,306);;
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, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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 ];;
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)(109,257)(110,258)(111,260)(112,259)
(113,253)(114,254)(115,256)(116,255)(117,261)(118,262)(119,264)(120,263)
(121,269)(122,270)(123,272)(124,271)(125,265)(126,266)(127,268)(128,267)
(129,273)(130,274)(131,276)(132,275)(133,281)(134,282)(135,284)(136,283)
(137,277)(138,278)(139,280)(140,279)(141,285)(142,286)(143,288)(144,287)
(145,221)(146,222)(147,224)(148,223)(149,217)(150,218)(151,220)(152,219)
(153,225)(154,226)(155,228)(156,227)(157,233)(158,234)(159,236)(160,235)
(161,229)(162,230)(163,232)(164,231)(165,237)(166,238)(167,240)(168,239)
(169,245)(170,246)(171,248)(172,247)(173,241)(174,242)(175,244)(176,243)
(177,249)(178,250)(179,252)(180,251)(181,293)(182,294)(183,296)(184,295)
(185,289)(186,290)(187,292)(188,291)(189,297)(190,298)(191,300)(192,299)
(193,305)(194,306)(195,308)(196,307)(197,301)(198,302)(199,304)(200,303)
(201,309)(202,310)(203,312)(204,311)(205,317)(206,318)(207,320)(208,319)
(209,313)(210,314)(211,316)(212,315)(213,321)(214,322)(215,324)(216,323);
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)(217,221)(218,224)(219,223)
(220,222)(226,228)(230,232)(233,237)(234,240)(235,239)(236,238)(241,249)
(242,252)(243,251)(244,250)(246,248)(253,321)(254,324)(255,323)(256,322)
(257,317)(258,320)(259,319)(260,318)(261,313)(262,316)(263,315)(264,314)
(265,293)(266,296)(267,295)(268,294)(269,289)(270,292)(271,291)(272,290)
(273,297)(274,300)(275,299)(276,298)(277,301)(278,304)(279,303)(280,302)
(281,309)(282,312)(283,311)(284,310)(285,305)(286,308)(287,307)(288,306);
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*s2*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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 >;

References : None.
to this polytope