Polytope of Type {9,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,12}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1789)
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 : {9,12}*432, {3,12}*432
   4-fold quotients : {9,6}*324a
   9-fold quotients : {9,4}*144, {3,12}*144
   12-fold quotients : {9,6}*108, {3,6}*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)(  7,  8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)
( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 37, 73)
( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 81)
( 46, 82)( 47, 84)( 48, 83)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)
( 54,102)( 55,104)( 56,103)( 57,105)( 58,106)( 59,108)( 60,107)( 61, 85)
( 62, 86)( 63, 88)( 64, 87)( 65, 89)( 66, 90)( 67, 92)( 68, 91)( 69, 93)
( 70, 94)( 71, 96)( 72, 95)(109,241)(110,242)(111,244)(112,243)(113,245)
(114,246)(115,248)(116,247)(117,249)(118,250)(119,252)(120,251)(121,229)
(122,230)(123,232)(124,231)(125,233)(126,234)(127,236)(128,235)(129,237)
(130,238)(131,240)(132,239)(133,217)(134,218)(135,220)(136,219)(137,221)
(138,222)(139,224)(140,223)(141,225)(142,226)(143,228)(144,227)(145,313)
(146,314)(147,316)(148,315)(149,317)(150,318)(151,320)(152,319)(153,321)
(154,322)(155,324)(156,323)(157,301)(158,302)(159,304)(160,303)(161,305)
(162,306)(163,308)(164,307)(165,309)(166,310)(167,312)(168,311)(169,289)
(170,290)(171,292)(172,291)(173,293)(174,294)(175,296)(176,295)(177,297)
(178,298)(179,300)(180,299)(181,277)(182,278)(183,280)(184,279)(185,281)
(186,282)(187,284)(188,283)(189,285)(190,286)(191,288)(192,287)(193,265)
(194,266)(195,268)(196,267)(197,269)(198,270)(199,272)(200,271)(201,273)
(202,274)(203,276)(204,275)(205,253)(206,254)(207,256)(208,255)(209,257)
(210,258)(211,260)(212,259)(213,261)(214,262)(215,264)(216,263);;
s1 := (  1,257)(  2,259)(  3,258)(  4,260)(  5,261)(  6,263)(  7,262)(  8,264)
(  9,253)( 10,255)( 11,254)( 12,256)( 13,281)( 14,283)( 15,282)( 16,284)
( 17,285)( 18,287)( 19,286)( 20,288)( 21,277)( 22,279)( 23,278)( 24,280)
( 25,269)( 26,271)( 27,270)( 28,272)( 29,273)( 30,275)( 31,274)( 32,276)
( 33,265)( 34,267)( 35,266)( 36,268)( 37,225)( 38,227)( 39,226)( 40,228)
( 41,217)( 42,219)( 43,218)( 44,220)( 45,221)( 46,223)( 47,222)( 48,224)
( 49,249)( 50,251)( 51,250)( 52,252)( 53,241)( 54,243)( 55,242)( 56,244)
( 57,245)( 58,247)( 59,246)( 60,248)( 61,237)( 62,239)( 63,238)( 64,240)
( 65,229)( 66,231)( 67,230)( 68,232)( 69,233)( 70,235)( 71,234)( 72,236)
( 73,289)( 74,291)( 75,290)( 76,292)( 77,293)( 78,295)( 79,294)( 80,296)
( 81,297)( 82,299)( 83,298)( 84,300)( 85,313)( 86,315)( 87,314)( 88,316)
( 89,317)( 90,319)( 91,318)( 92,320)( 93,321)( 94,323)( 95,322)( 96,324)
( 97,301)( 98,303)( 99,302)(100,304)(101,305)(102,307)(103,306)(104,308)
(105,309)(106,311)(107,310)(108,312)(109,149)(110,151)(111,150)(112,152)
(113,153)(114,155)(115,154)(116,156)(117,145)(118,147)(119,146)(120,148)
(121,173)(122,175)(123,174)(124,176)(125,177)(126,179)(127,178)(128,180)
(129,169)(130,171)(131,170)(132,172)(133,161)(134,163)(135,162)(136,164)
(137,165)(138,167)(139,166)(140,168)(141,157)(142,159)(143,158)(144,160)
(182,183)(186,187)(190,191)(193,205)(194,207)(195,206)(196,208)(197,209)
(198,211)(199,210)(200,212)(201,213)(202,215)(203,214)(204,216);;
s2 := (  1,  2)(  3,  4)(  5, 10)(  6,  9)(  7, 12)(  8, 11)( 13, 14)( 15, 16)
( 17, 22)( 18, 21)( 19, 24)( 20, 23)( 25, 26)( 27, 28)( 29, 34)( 30, 33)
( 31, 36)( 32, 35)( 37, 74)( 38, 73)( 39, 76)( 40, 75)( 41, 82)( 42, 81)
( 43, 84)( 44, 83)( 45, 78)( 46, 77)( 47, 80)( 48, 79)( 49, 86)( 50, 85)
( 51, 88)( 52, 87)( 53, 94)( 54, 93)( 55, 96)( 56, 95)( 57, 90)( 58, 89)
( 59, 92)( 60, 91)( 61, 98)( 62, 97)( 63,100)( 64, 99)( 65,106)( 66,105)
( 67,108)( 68,107)( 69,102)( 70,101)( 71,104)( 72,103)(109,110)(111,112)
(113,118)(114,117)(115,120)(116,119)(121,122)(123,124)(125,130)(126,129)
(127,132)(128,131)(133,134)(135,136)(137,142)(138,141)(139,144)(140,143)
(145,182)(146,181)(147,184)(148,183)(149,190)(150,189)(151,192)(152,191)
(153,186)(154,185)(155,188)(156,187)(157,194)(158,193)(159,196)(160,195)
(161,202)(162,201)(163,204)(164,203)(165,198)(166,197)(167,200)(168,199)
(169,206)(170,205)(171,208)(172,207)(173,214)(174,213)(175,216)(176,215)
(177,210)(178,209)(179,212)(180,211)(217,218)(219,220)(221,226)(222,225)
(223,228)(224,227)(229,230)(231,232)(233,238)(234,237)(235,240)(236,239)
(241,242)(243,244)(245,250)(246,249)(247,252)(248,251)(253,290)(254,289)
(255,292)(256,291)(257,298)(258,297)(259,300)(260,299)(261,294)(262,293)
(263,296)(264,295)(265,302)(266,301)(267,304)(268,303)(269,310)(270,309)
(271,312)(272,311)(273,306)(274,305)(275,308)(276,307)(277,314)(278,313)
(279,316)(280,315)(281,322)(282,321)(283,324)(284,323)(285,318)(286,317)
(287,320)(288,319);;
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, 
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)(  7,  8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)
( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)
( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)( 44, 79)
( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 97)( 50, 98)( 51,100)( 52, 99)
( 53,101)( 54,102)( 55,104)( 56,103)( 57,105)( 58,106)( 59,108)( 60,107)
( 61, 85)( 62, 86)( 63, 88)( 64, 87)( 65, 89)( 66, 90)( 67, 92)( 68, 91)
( 69, 93)( 70, 94)( 71, 96)( 72, 95)(109,241)(110,242)(111,244)(112,243)
(113,245)(114,246)(115,248)(116,247)(117,249)(118,250)(119,252)(120,251)
(121,229)(122,230)(123,232)(124,231)(125,233)(126,234)(127,236)(128,235)
(129,237)(130,238)(131,240)(132,239)(133,217)(134,218)(135,220)(136,219)
(137,221)(138,222)(139,224)(140,223)(141,225)(142,226)(143,228)(144,227)
(145,313)(146,314)(147,316)(148,315)(149,317)(150,318)(151,320)(152,319)
(153,321)(154,322)(155,324)(156,323)(157,301)(158,302)(159,304)(160,303)
(161,305)(162,306)(163,308)(164,307)(165,309)(166,310)(167,312)(168,311)
(169,289)(170,290)(171,292)(172,291)(173,293)(174,294)(175,296)(176,295)
(177,297)(178,298)(179,300)(180,299)(181,277)(182,278)(183,280)(184,279)
(185,281)(186,282)(187,284)(188,283)(189,285)(190,286)(191,288)(192,287)
(193,265)(194,266)(195,268)(196,267)(197,269)(198,270)(199,272)(200,271)
(201,273)(202,274)(203,276)(204,275)(205,253)(206,254)(207,256)(208,255)
(209,257)(210,258)(211,260)(212,259)(213,261)(214,262)(215,264)(216,263);
s1 := Sym(324)!(  1,257)(  2,259)(  3,258)(  4,260)(  5,261)(  6,263)(  7,262)
(  8,264)(  9,253)( 10,255)( 11,254)( 12,256)( 13,281)( 14,283)( 15,282)
( 16,284)( 17,285)( 18,287)( 19,286)( 20,288)( 21,277)( 22,279)( 23,278)
( 24,280)( 25,269)( 26,271)( 27,270)( 28,272)( 29,273)( 30,275)( 31,274)
( 32,276)( 33,265)( 34,267)( 35,266)( 36,268)( 37,225)( 38,227)( 39,226)
( 40,228)( 41,217)( 42,219)( 43,218)( 44,220)( 45,221)( 46,223)( 47,222)
( 48,224)( 49,249)( 50,251)( 51,250)( 52,252)( 53,241)( 54,243)( 55,242)
( 56,244)( 57,245)( 58,247)( 59,246)( 60,248)( 61,237)( 62,239)( 63,238)
( 64,240)( 65,229)( 66,231)( 67,230)( 68,232)( 69,233)( 70,235)( 71,234)
( 72,236)( 73,289)( 74,291)( 75,290)( 76,292)( 77,293)( 78,295)( 79,294)
( 80,296)( 81,297)( 82,299)( 83,298)( 84,300)( 85,313)( 86,315)( 87,314)
( 88,316)( 89,317)( 90,319)( 91,318)( 92,320)( 93,321)( 94,323)( 95,322)
( 96,324)( 97,301)( 98,303)( 99,302)(100,304)(101,305)(102,307)(103,306)
(104,308)(105,309)(106,311)(107,310)(108,312)(109,149)(110,151)(111,150)
(112,152)(113,153)(114,155)(115,154)(116,156)(117,145)(118,147)(119,146)
(120,148)(121,173)(122,175)(123,174)(124,176)(125,177)(126,179)(127,178)
(128,180)(129,169)(130,171)(131,170)(132,172)(133,161)(134,163)(135,162)
(136,164)(137,165)(138,167)(139,166)(140,168)(141,157)(142,159)(143,158)
(144,160)(182,183)(186,187)(190,191)(193,205)(194,207)(195,206)(196,208)
(197,209)(198,211)(199,210)(200,212)(201,213)(202,215)(203,214)(204,216);
s2 := Sym(324)!(  1,  2)(  3,  4)(  5, 10)(  6,  9)(  7, 12)(  8, 11)( 13, 14)
( 15, 16)( 17, 22)( 18, 21)( 19, 24)( 20, 23)( 25, 26)( 27, 28)( 29, 34)
( 30, 33)( 31, 36)( 32, 35)( 37, 74)( 38, 73)( 39, 76)( 40, 75)( 41, 82)
( 42, 81)( 43, 84)( 44, 83)( 45, 78)( 46, 77)( 47, 80)( 48, 79)( 49, 86)
( 50, 85)( 51, 88)( 52, 87)( 53, 94)( 54, 93)( 55, 96)( 56, 95)( 57, 90)
( 58, 89)( 59, 92)( 60, 91)( 61, 98)( 62, 97)( 63,100)( 64, 99)( 65,106)
( 66,105)( 67,108)( 68,107)( 69,102)( 70,101)( 71,104)( 72,103)(109,110)
(111,112)(113,118)(114,117)(115,120)(116,119)(121,122)(123,124)(125,130)
(126,129)(127,132)(128,131)(133,134)(135,136)(137,142)(138,141)(139,144)
(140,143)(145,182)(146,181)(147,184)(148,183)(149,190)(150,189)(151,192)
(152,191)(153,186)(154,185)(155,188)(156,187)(157,194)(158,193)(159,196)
(160,195)(161,202)(162,201)(163,204)(164,203)(165,198)(166,197)(167,200)
(168,199)(169,206)(170,205)(171,208)(172,207)(173,214)(174,213)(175,216)
(176,215)(177,210)(178,209)(179,212)(180,211)(217,218)(219,220)(221,226)
(222,225)(223,228)(224,227)(229,230)(231,232)(233,238)(234,237)(235,240)
(236,239)(241,242)(243,244)(245,250)(246,249)(247,252)(248,251)(253,290)
(254,289)(255,292)(256,291)(257,298)(258,297)(259,300)(260,299)(261,294)
(262,293)(263,296)(264,295)(265,302)(266,301)(267,304)(268,303)(269,310)
(270,309)(271,312)(272,311)(273,306)(274,305)(275,308)(276,307)(277,314)
(278,313)(279,316)(280,315)(281,322)(282,321)(283,324)(284,323)(285,318)
(286,317)(287,320)(288,319);
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, 
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