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Polytope of Type {9,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,4}*1296b
if this polytope has a name.
Group : SmallGroup(1296,917)
Rank : 4
Schlafli Type : {9,6,4}
Number of vertices, edges, etc : 27, 81, 36, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {9,6,2}*648c
   3-fold quotients : {3,6,4}*432a
   6-fold quotients : {3,6,2}*216
   9-fold quotients : {3,6,4}*144
   18-fold quotients : {3,6,2}*72
   27-fold quotients : {3,2,4}*48
   54-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  3)(  4,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)
( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 60)
( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 73)( 38, 75)( 39, 74)
( 40, 78)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 81)( 46, 64)( 47, 66)
( 48, 65)( 49, 69)( 50, 68)( 51, 67)( 52, 71)( 53, 70)( 54, 72)( 83, 84)
( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)
( 97,107)( 98,106)( 99,108)(109,136)(110,138)(111,137)(112,141)(113,140)
(114,139)(115,143)(116,142)(117,144)(118,154)(119,156)(120,155)(121,159)
(122,158)(123,157)(124,161)(125,160)(126,162)(127,145)(128,147)(129,146)
(130,150)(131,149)(132,148)(133,152)(134,151)(135,153)(164,165)(166,168)
(169,170)(172,181)(173,183)(174,182)(175,186)(176,185)(177,184)(178,188)
(179,187)(180,189)(190,217)(191,219)(192,218)(193,222)(194,221)(195,220)
(196,224)(197,223)(198,225)(199,235)(200,237)(201,236)(202,240)(203,239)
(204,238)(205,242)(206,241)(207,243)(208,226)(209,228)(210,227)(211,231)
(212,230)(213,229)(214,233)(215,232)(216,234)(245,246)(247,249)(250,251)
(253,262)(254,264)(255,263)(256,267)(257,266)(258,265)(259,269)(260,268)
(261,270)(271,298)(272,300)(273,299)(274,303)(275,302)(276,301)(277,305)
(278,304)(279,306)(280,316)(281,318)(282,317)(283,321)(284,320)(285,319)
(286,323)(287,322)(288,324)(289,307)(290,309)(291,308)(292,312)(293,311)
(294,310)(295,314)(296,313)(297,315);;
s1 := (  1, 67)(  2, 69)(  3, 68)(  4, 70)(  5, 72)(  6, 71)(  7, 64)(  8, 66)
(  9, 65)( 10, 61)( 11, 63)( 12, 62)( 13, 55)( 14, 57)( 15, 56)( 16, 58)
( 17, 60)( 18, 59)( 19, 75)( 20, 74)( 21, 73)( 22, 78)( 23, 77)( 24, 76)
( 25, 81)( 26, 80)( 27, 79)( 28, 40)( 29, 42)( 30, 41)( 31, 43)( 32, 45)
( 33, 44)( 34, 37)( 35, 39)( 36, 38)( 46, 48)( 49, 51)( 52, 54)( 82,148)
( 83,150)( 84,149)( 85,151)( 86,153)( 87,152)( 88,145)( 89,147)( 90,146)
( 91,142)( 92,144)( 93,143)( 94,136)( 95,138)( 96,137)( 97,139)( 98,141)
( 99,140)(100,156)(101,155)(102,154)(103,159)(104,158)(105,157)(106,162)
(107,161)(108,160)(109,121)(110,123)(111,122)(112,124)(113,126)(114,125)
(115,118)(116,120)(117,119)(127,129)(130,132)(133,135)(163,229)(164,231)
(165,230)(166,232)(167,234)(168,233)(169,226)(170,228)(171,227)(172,223)
(173,225)(174,224)(175,217)(176,219)(177,218)(178,220)(179,222)(180,221)
(181,237)(182,236)(183,235)(184,240)(185,239)(186,238)(187,243)(188,242)
(189,241)(190,202)(191,204)(192,203)(193,205)(194,207)(195,206)(196,199)
(197,201)(198,200)(208,210)(211,213)(214,216)(244,310)(245,312)(246,311)
(247,313)(248,315)(249,314)(250,307)(251,309)(252,308)(253,304)(254,306)
(255,305)(256,298)(257,300)(258,299)(259,301)(260,303)(261,302)(262,318)
(263,317)(264,316)(265,321)(266,320)(267,319)(268,324)(269,323)(270,322)
(271,283)(272,285)(273,284)(274,286)(275,288)(276,287)(277,280)(278,282)
(279,281)(289,291)(292,294)(295,297);;
s2 := (  4,  8)(  5,  9)(  6,  7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)
( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)
( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)
( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)
( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 85, 89)( 86, 90)( 87, 88)( 91,100)
( 92,101)( 93,102)( 94,107)( 95,108)( 96,106)( 97,105)( 98,103)( 99,104)
(112,116)(113,117)(114,115)(118,127)(119,128)(120,129)(121,134)(122,135)
(123,133)(124,132)(125,130)(126,131)(139,143)(140,144)(141,142)(145,154)
(146,155)(147,156)(148,161)(149,162)(150,160)(151,159)(152,157)(153,158)
(163,244)(164,245)(165,246)(166,251)(167,252)(168,250)(169,249)(170,247)
(171,248)(172,262)(173,263)(174,264)(175,269)(176,270)(177,268)(178,267)
(179,265)(180,266)(181,253)(182,254)(183,255)(184,260)(185,261)(186,259)
(187,258)(188,256)(189,257)(190,271)(191,272)(192,273)(193,278)(194,279)
(195,277)(196,276)(197,274)(198,275)(199,289)(200,290)(201,291)(202,296)
(203,297)(204,295)(205,294)(206,292)(207,293)(208,280)(209,281)(210,282)
(211,287)(212,288)(213,286)(214,285)(215,283)(216,284)(217,298)(218,299)
(219,300)(220,305)(221,306)(222,304)(223,303)(224,301)(225,302)(226,316)
(227,317)(228,318)(229,323)(230,324)(231,322)(232,321)(233,319)(234,320)
(235,307)(236,308)(237,309)(238,314)(239,315)(240,313)(241,312)(242,310)
(243,311);;
s3 := (  1,163)(  2,164)(  3,165)(  4,166)(  5,167)(  6,168)(  7,169)(  8,170)
(  9,171)( 10,172)( 11,173)( 12,174)( 13,175)( 14,176)( 15,177)( 16,178)
( 17,179)( 18,180)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)( 24,186)
( 25,187)( 26,188)( 27,189)( 28,190)( 29,191)( 30,192)( 31,193)( 32,194)
( 33,195)( 34,196)( 35,197)( 36,198)( 37,199)( 38,200)( 39,201)( 40,202)
( 41,203)( 42,204)( 43,205)( 44,206)( 45,207)( 46,208)( 47,209)( 48,210)
( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,217)( 56,218)
( 57,219)( 58,220)( 59,221)( 60,222)( 61,223)( 62,224)( 63,225)( 64,226)
( 65,227)( 66,228)( 67,229)( 68,230)( 69,231)( 70,232)( 71,233)( 72,234)
( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)( 80,242)
( 81,243)( 82,244)( 83,245)( 84,246)( 85,247)( 86,248)( 87,249)( 88,250)
( 89,251)( 90,252)( 91,253)( 92,254)( 93,255)( 94,256)( 95,257)( 96,258)
( 97,259)( 98,260)( 99,261)(100,262)(101,263)(102,264)(103,265)(104,266)
(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)(112,274)
(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)(120,282)
(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)(128,290)
(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,298)
(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)(144,306)
(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)(152,314)
(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)
(161,323)(162,324);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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)!(  2,  3)(  4,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)
( 14, 23)( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 55)( 29, 57)( 30, 56)
( 31, 60)( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 73)( 38, 75)
( 39, 74)( 40, 78)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 81)( 46, 64)
( 47, 66)( 48, 65)( 49, 69)( 50, 68)( 51, 67)( 52, 71)( 53, 70)( 54, 72)
( 83, 84)( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)
( 96,103)( 97,107)( 98,106)( 99,108)(109,136)(110,138)(111,137)(112,141)
(113,140)(114,139)(115,143)(116,142)(117,144)(118,154)(119,156)(120,155)
(121,159)(122,158)(123,157)(124,161)(125,160)(126,162)(127,145)(128,147)
(129,146)(130,150)(131,149)(132,148)(133,152)(134,151)(135,153)(164,165)
(166,168)(169,170)(172,181)(173,183)(174,182)(175,186)(176,185)(177,184)
(178,188)(179,187)(180,189)(190,217)(191,219)(192,218)(193,222)(194,221)
(195,220)(196,224)(197,223)(198,225)(199,235)(200,237)(201,236)(202,240)
(203,239)(204,238)(205,242)(206,241)(207,243)(208,226)(209,228)(210,227)
(211,231)(212,230)(213,229)(214,233)(215,232)(216,234)(245,246)(247,249)
(250,251)(253,262)(254,264)(255,263)(256,267)(257,266)(258,265)(259,269)
(260,268)(261,270)(271,298)(272,300)(273,299)(274,303)(275,302)(276,301)
(277,305)(278,304)(279,306)(280,316)(281,318)(282,317)(283,321)(284,320)
(285,319)(286,323)(287,322)(288,324)(289,307)(290,309)(291,308)(292,312)
(293,311)(294,310)(295,314)(296,313)(297,315);
s1 := Sym(324)!(  1, 67)(  2, 69)(  3, 68)(  4, 70)(  5, 72)(  6, 71)(  7, 64)
(  8, 66)(  9, 65)( 10, 61)( 11, 63)( 12, 62)( 13, 55)( 14, 57)( 15, 56)
( 16, 58)( 17, 60)( 18, 59)( 19, 75)( 20, 74)( 21, 73)( 22, 78)( 23, 77)
( 24, 76)( 25, 81)( 26, 80)( 27, 79)( 28, 40)( 29, 42)( 30, 41)( 31, 43)
( 32, 45)( 33, 44)( 34, 37)( 35, 39)( 36, 38)( 46, 48)( 49, 51)( 52, 54)
( 82,148)( 83,150)( 84,149)( 85,151)( 86,153)( 87,152)( 88,145)( 89,147)
( 90,146)( 91,142)( 92,144)( 93,143)( 94,136)( 95,138)( 96,137)( 97,139)
( 98,141)( 99,140)(100,156)(101,155)(102,154)(103,159)(104,158)(105,157)
(106,162)(107,161)(108,160)(109,121)(110,123)(111,122)(112,124)(113,126)
(114,125)(115,118)(116,120)(117,119)(127,129)(130,132)(133,135)(163,229)
(164,231)(165,230)(166,232)(167,234)(168,233)(169,226)(170,228)(171,227)
(172,223)(173,225)(174,224)(175,217)(176,219)(177,218)(178,220)(179,222)
(180,221)(181,237)(182,236)(183,235)(184,240)(185,239)(186,238)(187,243)
(188,242)(189,241)(190,202)(191,204)(192,203)(193,205)(194,207)(195,206)
(196,199)(197,201)(198,200)(208,210)(211,213)(214,216)(244,310)(245,312)
(246,311)(247,313)(248,315)(249,314)(250,307)(251,309)(252,308)(253,304)
(254,306)(255,305)(256,298)(257,300)(258,299)(259,301)(260,303)(261,302)
(262,318)(263,317)(264,316)(265,321)(266,320)(267,319)(268,324)(269,323)
(270,322)(271,283)(272,285)(273,284)(274,286)(275,288)(276,287)(277,280)
(278,282)(279,281)(289,291)(292,294)(295,297);
s2 := Sym(324)!(  4,  8)(  5,  9)(  6,  7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)
( 14, 27)( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)
( 37, 46)( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)
( 45, 50)( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)
( 68, 81)( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 85, 89)( 86, 90)( 87, 88)
( 91,100)( 92,101)( 93,102)( 94,107)( 95,108)( 96,106)( 97,105)( 98,103)
( 99,104)(112,116)(113,117)(114,115)(118,127)(119,128)(120,129)(121,134)
(122,135)(123,133)(124,132)(125,130)(126,131)(139,143)(140,144)(141,142)
(145,154)(146,155)(147,156)(148,161)(149,162)(150,160)(151,159)(152,157)
(153,158)(163,244)(164,245)(165,246)(166,251)(167,252)(168,250)(169,249)
(170,247)(171,248)(172,262)(173,263)(174,264)(175,269)(176,270)(177,268)
(178,267)(179,265)(180,266)(181,253)(182,254)(183,255)(184,260)(185,261)
(186,259)(187,258)(188,256)(189,257)(190,271)(191,272)(192,273)(193,278)
(194,279)(195,277)(196,276)(197,274)(198,275)(199,289)(200,290)(201,291)
(202,296)(203,297)(204,295)(205,294)(206,292)(207,293)(208,280)(209,281)
(210,282)(211,287)(212,288)(213,286)(214,285)(215,283)(216,284)(217,298)
(218,299)(219,300)(220,305)(221,306)(222,304)(223,303)(224,301)(225,302)
(226,316)(227,317)(228,318)(229,323)(230,324)(231,322)(232,321)(233,319)
(234,320)(235,307)(236,308)(237,309)(238,314)(239,315)(240,313)(241,312)
(242,310)(243,311);
s3 := Sym(324)!(  1,163)(  2,164)(  3,165)(  4,166)(  5,167)(  6,168)(  7,169)
(  8,170)(  9,171)( 10,172)( 11,173)( 12,174)( 13,175)( 14,176)( 15,177)
( 16,178)( 17,179)( 18,180)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)
( 24,186)( 25,187)( 26,188)( 27,189)( 28,190)( 29,191)( 30,192)( 31,193)
( 32,194)( 33,195)( 34,196)( 35,197)( 36,198)( 37,199)( 38,200)( 39,201)
( 40,202)( 41,203)( 42,204)( 43,205)( 44,206)( 45,207)( 46,208)( 47,209)
( 48,210)( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,217)
( 56,218)( 57,219)( 58,220)( 59,221)( 60,222)( 61,223)( 62,224)( 63,225)
( 64,226)( 65,227)( 66,228)( 67,229)( 68,230)( 69,231)( 70,232)( 71,233)
( 72,234)( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)
( 80,242)( 81,243)( 82,244)( 83,245)( 84,246)( 85,247)( 86,248)( 87,249)
( 88,250)( 89,251)( 90,252)( 91,253)( 92,254)( 93,255)( 94,256)( 95,257)
( 96,258)( 97,259)( 98,260)( 99,261)(100,262)(101,263)(102,264)(103,265)
(104,266)(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)
(112,274)(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)
(120,282)(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)
(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)
(136,298)(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)
(144,306)(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)
(152,314)(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)
(160,322)(161,323)(162,324);
poly := sub<Sym(324)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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