Polytope of Type {9,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,4}*1296c
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}*648b
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 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 55)( 29, 57)( 30, 56)( 31, 61)
( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)( 39, 65)
( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)( 47, 75)
( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)
(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)
(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)
(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(163,244)(164,246)
(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)(172,253)
(173,255)(174,254)(175,259)(176,261)(177,260)(178,256)(179,258)(180,257)
(181,262)(182,264)(183,263)(184,268)(185,270)(186,269)(187,265)(188,267)
(189,266)(190,298)(191,300)(192,299)(193,304)(194,306)(195,305)(196,301)
(197,303)(198,302)(199,307)(200,309)(201,308)(202,313)(203,315)(204,314)
(205,310)(206,312)(207,311)(208,316)(209,318)(210,317)(211,322)(212,324)
(213,323)(214,319)(215,321)(216,320)(217,271)(218,273)(219,272)(220,277)
(221,279)(222,278)(223,274)(224,276)(225,275)(226,280)(227,282)(228,281)
(229,286)(230,288)(231,287)(232,283)(233,285)(234,284)(235,289)(236,291)
(237,290)(238,295)(239,297)(240,296)(241,292)(242,294)(243,293);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1,
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)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 55)( 29, 57)( 30, 56)
( 31, 61)( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)
( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)
( 47, 75)( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)
(113,144)(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)
(121,151)(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)
(129,155)(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(163,244)
(164,246)(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)
(172,253)(173,255)(174,254)(175,259)(176,261)(177,260)(178,256)(179,258)
(180,257)(181,262)(182,264)(183,263)(184,268)(185,270)(186,269)(187,265)
(188,267)(189,266)(190,298)(191,300)(192,299)(193,304)(194,306)(195,305)
(196,301)(197,303)(198,302)(199,307)(200,309)(201,308)(202,313)(203,315)
(204,314)(205,310)(206,312)(207,311)(208,316)(209,318)(210,317)(211,322)
(212,324)(213,323)(214,319)(215,321)(216,320)(217,271)(218,273)(219,272)
(220,277)(221,279)(222,278)(223,274)(224,276)(225,275)(226,280)(227,282)
(228,281)(229,286)(230,288)(231,287)(232,283)(233,285)(234,284)(235,289)
(236,291)(237,290)(238,295)(239,297)(240,296)(241,292)(242,294)(243,293);
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1,
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