Polytope of Type {6,9,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9,4}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1789)
Rank : 4
Schlafli Type : {6,9,4}
Number of vertices, edges, etc : 18, 81, 54, 4
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-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) :
3-fold quotients : {6,9,4}*432, {6,3,4}*432
9-fold quotients : {2,9,4}*144, {6,3,4}*144
27-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 37, 73)( 38, 74)( 39, 75)( 40, 76)
( 41, 81)( 42, 82)( 43, 83)( 44, 84)( 45, 77)( 46, 78)( 47, 79)( 48, 80)
( 49, 85)( 50, 86)( 51, 87)( 52, 88)( 53, 93)( 54, 94)( 55, 95)( 56, 96)
( 57, 89)( 58, 90)( 59, 91)( 60, 92)( 61, 97)( 62, 98)( 63, 99)( 64,100)
( 65,105)( 66,106)( 67,107)( 68,108)( 69,101)( 70,102)( 71,103)( 72,104)
(113,117)(114,118)(115,119)(116,120)(125,129)(126,130)(127,131)(128,132)
(137,141)(138,142)(139,143)(140,144)(145,181)(146,182)(147,183)(148,184)
(149,189)(150,190)(151,191)(152,192)(153,185)(154,186)(155,187)(156,188)
(157,193)(158,194)(159,195)(160,196)(161,201)(162,202)(163,203)(164,204)
(165,197)(166,198)(167,199)(168,200)(169,205)(170,206)(171,207)(172,208)
(173,213)(174,214)(175,215)(176,216)(177,209)(178,210)(179,211)(180,212)
(221,225)(222,226)(223,227)(224,228)(233,237)(234,238)(235,239)(236,240)
(245,249)(246,250)(247,251)(248,252)(253,289)(254,290)(255,291)(256,292)
(257,297)(258,298)(259,299)(260,300)(261,293)(262,294)(263,295)(264,296)
(265,301)(266,302)(267,303)(268,304)(269,309)(270,310)(271,311)(272,312)
(273,305)(274,306)(275,307)(276,308)(277,313)(278,314)(279,315)(280,316)
(281,321)(282,322)(283,323)(284,324)(285,317)(286,318)(287,319)(288,320);;
s1 := ( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 41)( 6, 42)( 7, 44)( 8, 43)
( 9, 45)( 10, 46)( 11, 48)( 12, 47)( 13, 61)( 14, 62)( 15, 64)( 16, 63)
( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 69)( 22, 70)( 23, 72)( 24, 71)
( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 53)( 30, 54)( 31, 56)( 32, 55)
( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 75, 76)( 79, 80)( 83, 84)( 85, 97)
( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)( 91,104)( 92,103)( 93,105)
( 94,106)( 95,108)( 96,107)(109,277)(110,278)(111,280)(112,279)(113,281)
(114,282)(115,284)(116,283)(117,285)(118,286)(119,288)(120,287)(121,265)
(122,266)(123,268)(124,267)(125,269)(126,270)(127,272)(128,271)(129,273)
(130,274)(131,276)(132,275)(133,253)(134,254)(135,256)(136,255)(137,257)
(138,258)(139,260)(140,259)(141,261)(142,262)(143,264)(144,263)(145,241)
(146,242)(147,244)(148,243)(149,245)(150,246)(151,248)(152,247)(153,249)
(154,250)(155,252)(156,251)(157,229)(158,230)(159,232)(160,231)(161,233)
(162,234)(163,236)(164,235)(165,237)(166,238)(167,240)(168,239)(169,217)
(170,218)(171,220)(172,219)(173,221)(174,222)(175,224)(176,223)(177,225)
(178,226)(179,228)(180,227)(181,313)(182,314)(183,316)(184,315)(185,317)
(186,318)(187,320)(188,319)(189,321)(190,322)(191,324)(192,323)(193,301)
(194,302)(195,304)(196,303)(197,305)(198,306)(199,308)(200,307)(201,309)
(202,310)(203,312)(204,311)(205,289)(206,290)(207,292)(208,291)(209,293)
(210,294)(211,296)(212,295)(213,297)(214,298)(215,300)(216,299);;
s2 := ( 1,109)( 2,112)( 3,111)( 4,110)( 5,113)( 6,116)( 7,115)( 8,114)
( 9,117)( 10,120)( 11,119)( 12,118)( 13,133)( 14,136)( 15,135)( 16,134)
( 17,137)( 18,140)( 19,139)( 20,138)( 21,141)( 22,144)( 23,143)( 24,142)
( 25,121)( 26,124)( 27,123)( 28,122)( 29,125)( 30,128)( 31,127)( 32,126)
( 33,129)( 34,132)( 35,131)( 36,130)( 37,189)( 38,192)( 39,191)( 40,190)
( 41,181)( 42,184)( 43,183)( 44,182)( 45,185)( 46,188)( 47,187)( 48,186)
( 49,213)( 50,216)( 51,215)( 52,214)( 53,205)( 54,208)( 55,207)( 56,206)
( 57,209)( 58,212)( 59,211)( 60,210)( 61,201)( 62,204)( 63,203)( 64,202)
( 65,193)( 66,196)( 67,195)( 68,194)( 69,197)( 70,200)( 71,199)( 72,198)
( 73,149)( 74,152)( 75,151)( 76,150)( 77,153)( 78,156)( 79,155)( 80,154)
( 81,145)( 82,148)( 83,147)( 84,146)( 85,173)( 86,176)( 87,175)( 88,174)
( 89,177)( 90,180)( 91,179)( 92,178)( 93,169)( 94,172)( 95,171)( 96,170)
( 97,161)( 98,164)( 99,163)(100,162)(101,165)(102,168)(103,167)(104,166)
(105,157)(106,160)(107,159)(108,158)(217,241)(218,244)(219,243)(220,242)
(221,245)(222,248)(223,247)(224,246)(225,249)(226,252)(227,251)(228,250)
(230,232)(234,236)(238,240)(253,321)(254,324)(255,323)(256,322)(257,313)
(258,316)(259,315)(260,314)(261,317)(262,320)(263,319)(264,318)(265,309)
(266,312)(267,311)(268,310)(269,301)(270,304)(271,303)(272,302)(273,305)
(274,308)(275,307)(276,306)(277,297)(278,300)(279,299)(280,298)(281,289)
(282,292)(283,291)(284,290)(285,293)(286,296)(287,295)(288,294);;
s3 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)
(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)
(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)
(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)
(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)
(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)
(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)
(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304)
(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)(319,320)
(321,322)(323,324);;
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, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s2*s1*s3*s2*s3*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1,
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)!( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 17, 21)( 18, 22)( 19, 23)
( 20, 24)( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 37, 73)( 38, 74)( 39, 75)
( 40, 76)( 41, 81)( 42, 82)( 43, 83)( 44, 84)( 45, 77)( 46, 78)( 47, 79)
( 48, 80)( 49, 85)( 50, 86)( 51, 87)( 52, 88)( 53, 93)( 54, 94)( 55, 95)
( 56, 96)( 57, 89)( 58, 90)( 59, 91)( 60, 92)( 61, 97)( 62, 98)( 63, 99)
( 64,100)( 65,105)( 66,106)( 67,107)( 68,108)( 69,101)( 70,102)( 71,103)
( 72,104)(113,117)(114,118)(115,119)(116,120)(125,129)(126,130)(127,131)
(128,132)(137,141)(138,142)(139,143)(140,144)(145,181)(146,182)(147,183)
(148,184)(149,189)(150,190)(151,191)(152,192)(153,185)(154,186)(155,187)
(156,188)(157,193)(158,194)(159,195)(160,196)(161,201)(162,202)(163,203)
(164,204)(165,197)(166,198)(167,199)(168,200)(169,205)(170,206)(171,207)
(172,208)(173,213)(174,214)(175,215)(176,216)(177,209)(178,210)(179,211)
(180,212)(221,225)(222,226)(223,227)(224,228)(233,237)(234,238)(235,239)
(236,240)(245,249)(246,250)(247,251)(248,252)(253,289)(254,290)(255,291)
(256,292)(257,297)(258,298)(259,299)(260,300)(261,293)(262,294)(263,295)
(264,296)(265,301)(266,302)(267,303)(268,304)(269,309)(270,310)(271,311)
(272,312)(273,305)(274,306)(275,307)(276,308)(277,313)(278,314)(279,315)
(280,316)(281,321)(282,322)(283,323)(284,324)(285,317)(286,318)(287,319)
(288,320);
s1 := Sym(324)!( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 41)( 6, 42)( 7, 44)
( 8, 43)( 9, 45)( 10, 46)( 11, 48)( 12, 47)( 13, 61)( 14, 62)( 15, 64)
( 16, 63)( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 69)( 22, 70)( 23, 72)
( 24, 71)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 53)( 30, 54)( 31, 56)
( 32, 55)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 75, 76)( 79, 80)( 83, 84)
( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)( 91,104)( 92,103)
( 93,105)( 94,106)( 95,108)( 96,107)(109,277)(110,278)(111,280)(112,279)
(113,281)(114,282)(115,284)(116,283)(117,285)(118,286)(119,288)(120,287)
(121,265)(122,266)(123,268)(124,267)(125,269)(126,270)(127,272)(128,271)
(129,273)(130,274)(131,276)(132,275)(133,253)(134,254)(135,256)(136,255)
(137,257)(138,258)(139,260)(140,259)(141,261)(142,262)(143,264)(144,263)
(145,241)(146,242)(147,244)(148,243)(149,245)(150,246)(151,248)(152,247)
(153,249)(154,250)(155,252)(156,251)(157,229)(158,230)(159,232)(160,231)
(161,233)(162,234)(163,236)(164,235)(165,237)(166,238)(167,240)(168,239)
(169,217)(170,218)(171,220)(172,219)(173,221)(174,222)(175,224)(176,223)
(177,225)(178,226)(179,228)(180,227)(181,313)(182,314)(183,316)(184,315)
(185,317)(186,318)(187,320)(188,319)(189,321)(190,322)(191,324)(192,323)
(193,301)(194,302)(195,304)(196,303)(197,305)(198,306)(199,308)(200,307)
(201,309)(202,310)(203,312)(204,311)(205,289)(206,290)(207,292)(208,291)
(209,293)(210,294)(211,296)(212,295)(213,297)(214,298)(215,300)(216,299);
s2 := Sym(324)!( 1,109)( 2,112)( 3,111)( 4,110)( 5,113)( 6,116)( 7,115)
( 8,114)( 9,117)( 10,120)( 11,119)( 12,118)( 13,133)( 14,136)( 15,135)
( 16,134)( 17,137)( 18,140)( 19,139)( 20,138)( 21,141)( 22,144)( 23,143)
( 24,142)( 25,121)( 26,124)( 27,123)( 28,122)( 29,125)( 30,128)( 31,127)
( 32,126)( 33,129)( 34,132)( 35,131)( 36,130)( 37,189)( 38,192)( 39,191)
( 40,190)( 41,181)( 42,184)( 43,183)( 44,182)( 45,185)( 46,188)( 47,187)
( 48,186)( 49,213)( 50,216)( 51,215)( 52,214)( 53,205)( 54,208)( 55,207)
( 56,206)( 57,209)( 58,212)( 59,211)( 60,210)( 61,201)( 62,204)( 63,203)
( 64,202)( 65,193)( 66,196)( 67,195)( 68,194)( 69,197)( 70,200)( 71,199)
( 72,198)( 73,149)( 74,152)( 75,151)( 76,150)( 77,153)( 78,156)( 79,155)
( 80,154)( 81,145)( 82,148)( 83,147)( 84,146)( 85,173)( 86,176)( 87,175)
( 88,174)( 89,177)( 90,180)( 91,179)( 92,178)( 93,169)( 94,172)( 95,171)
( 96,170)( 97,161)( 98,164)( 99,163)(100,162)(101,165)(102,168)(103,167)
(104,166)(105,157)(106,160)(107,159)(108,158)(217,241)(218,244)(219,243)
(220,242)(221,245)(222,248)(223,247)(224,246)(225,249)(226,252)(227,251)
(228,250)(230,232)(234,236)(238,240)(253,321)(254,324)(255,323)(256,322)
(257,313)(258,316)(259,315)(260,314)(261,317)(262,320)(263,319)(264,318)
(265,309)(266,312)(267,311)(268,310)(269,301)(270,304)(271,303)(272,302)
(273,305)(274,308)(275,307)(276,306)(277,297)(278,300)(279,299)(280,298)
(281,289)(282,292)(283,291)(284,290)(285,293)(286,296)(287,295)(288,294);
s3 := Sym(324)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)
(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)
(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)
(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)
(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)
(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)
(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)
(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)
(303,304)(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)
(319,320)(321,322)(323,324);
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,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope