Polytope of Type {9,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,12}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1787)
Rank : 3
Schlafli Type : {9,12}
Number of vertices, edges, etc : 54, 324, 72
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 : {3,12}*432
   4-fold quotients : {9,6}*324d
   9-fold quotients : {3,12}*144
   12-fold quotients : {3,6}*108
   27-fold quotients : {3,4}*48
   36-fold quotients : {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, 21)( 14, 22)( 15, 24)
( 16, 23)( 19, 20)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 35, 36)( 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, 93)( 50, 94)( 51, 96)( 52, 95)( 53, 89)
( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)( 60, 87)( 61,101)
( 62,102)( 63,104)( 64,103)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69,105)
( 70,106)( 71,108)( 72,107)(109,221)(110,222)(111,224)(112,223)(113,217)
(114,218)(115,220)(116,219)(117,225)(118,226)(119,228)(120,227)(121,229)
(122,230)(123,232)(124,231)(125,237)(126,238)(127,240)(128,239)(129,233)
(130,234)(131,236)(132,235)(133,249)(134,250)(135,252)(136,251)(137,245)
(138,246)(139,248)(140,247)(141,241)(142,242)(143,244)(144,243)(145,293)
(146,294)(147,296)(148,295)(149,289)(150,290)(151,292)(152,291)(153,297)
(154,298)(155,300)(156,299)(157,301)(158,302)(159,304)(160,303)(161,309)
(162,310)(163,312)(164,311)(165,305)(166,306)(167,308)(168,307)(169,321)
(170,322)(171,324)(172,323)(173,317)(174,318)(175,320)(176,319)(177,313)
(178,314)(179,316)(180,315)(181,257)(182,258)(183,260)(184,259)(185,253)
(186,254)(187,256)(188,255)(189,261)(190,262)(191,264)(192,263)(193,265)
(194,266)(195,268)(196,267)(197,273)(198,274)(199,276)(200,275)(201,269)
(202,270)(203,272)(204,271)(205,285)(206,286)(207,288)(208,287)(209,281)
(210,282)(211,284)(212,283)(213,277)(214,278)(215,280)(216,279);;
s1 := (  1,265)(  2,267)(  3,266)(  4,268)(  5,273)(  6,275)(  7,274)(  8,276)
(  9,269)( 10,271)( 11,270)( 12,272)( 13,277)( 14,279)( 15,278)( 16,280)
( 17,285)( 18,287)( 19,286)( 20,288)( 21,281)( 22,283)( 23,282)( 24,284)
( 25,253)( 26,255)( 27,254)( 28,256)( 29,261)( 30,263)( 31,262)( 32,264)
( 33,257)( 34,259)( 35,258)( 36,260)( 37,241)( 38,243)( 39,242)( 40,244)
( 41,249)( 42,251)( 43,250)( 44,252)( 45,245)( 46,247)( 47,246)( 48,248)
( 49,217)( 50,219)( 51,218)( 52,220)( 53,225)( 54,227)( 55,226)( 56,228)
( 57,221)( 58,223)( 59,222)( 60,224)( 61,229)( 62,231)( 63,230)( 64,232)
( 65,237)( 66,239)( 67,238)( 68,240)( 69,233)( 70,235)( 71,234)( 72,236)
( 73,297)( 74,299)( 75,298)( 76,300)( 77,293)( 78,295)( 79,294)( 80,296)
( 81,289)( 82,291)( 83,290)( 84,292)( 85,309)( 86,311)( 87,310)( 88,312)
( 89,305)( 90,307)( 91,306)( 92,308)( 93,301)( 94,303)( 95,302)( 96,304)
( 97,321)( 98,323)( 99,322)(100,324)(101,317)(102,319)(103,318)(104,320)
(105,313)(106,315)(107,314)(108,316)(109,157)(110,159)(111,158)(112,160)
(113,165)(114,167)(115,166)(116,168)(117,161)(118,163)(119,162)(120,164)
(121,169)(122,171)(123,170)(124,172)(125,177)(126,179)(127,178)(128,180)
(129,173)(130,175)(131,174)(132,176)(133,145)(134,147)(135,146)(136,148)
(137,153)(138,155)(139,154)(140,156)(141,149)(142,151)(143,150)(144,152)
(181,189)(182,191)(183,190)(184,192)(186,187)(193,201)(194,203)(195,202)
(196,204)(198,199)(205,213)(206,215)(207,214)(208,216)(210,211);;
s2 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 30)( 14, 29)
( 15, 32)( 16, 31)( 17, 34)( 18, 33)( 19, 36)( 20, 35)( 21, 26)( 22, 25)
( 23, 28)( 24, 27)( 37, 74)( 38, 73)( 39, 76)( 40, 75)( 41, 78)( 42, 77)
( 43, 80)( 44, 79)( 45, 82)( 46, 81)( 47, 84)( 48, 83)( 49,102)( 50,101)
( 51,104)( 52,103)( 53,106)( 54,105)( 55,108)( 56,107)( 57, 98)( 58, 97)
( 59,100)( 60, 99)( 61, 94)( 62, 93)( 63, 96)( 64, 95)( 65, 86)( 66, 85)
( 67, 88)( 68, 87)( 69, 90)( 70, 89)( 71, 92)( 72, 91)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,138)(122,137)(123,140)(124,139)
(125,142)(126,141)(127,144)(128,143)(129,134)(130,133)(131,136)(132,135)
(145,182)(146,181)(147,184)(148,183)(149,186)(150,185)(151,188)(152,187)
(153,190)(154,189)(155,192)(156,191)(157,210)(158,209)(159,212)(160,211)
(161,214)(162,213)(163,216)(164,215)(165,206)(166,205)(167,208)(168,207)
(169,202)(170,201)(171,204)(172,203)(173,194)(174,193)(175,196)(176,195)
(177,198)(178,197)(179,200)(180,199)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,246)(230,245)(231,248)(232,247)(233,250)(234,249)
(235,252)(236,251)(237,242)(238,241)(239,244)(240,243)(253,290)(254,289)
(255,292)(256,291)(257,294)(258,293)(259,296)(260,295)(261,298)(262,297)
(263,300)(264,299)(265,318)(266,317)(267,320)(268,319)(269,322)(270,321)
(271,324)(272,323)(273,314)(274,313)(275,316)(276,315)(277,310)(278,309)
(279,312)(280,311)(281,302)(282,301)(283,304)(284,303)(285,306)(286,305)
(287,308)(288,307);;
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, 
s1*s2*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope