Polytope of Type {4,9,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,9,6}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1787)
Rank : 4
Schlafli Type : {4,9,6}
Number of vertices, edges, etc : 4, 54, 81, 18
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 : {4,3,6}*432
9-fold quotients : {4,3,6}*144
27-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320)
(321,323)(322,324);;
s1 := ( 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);;
s2 := ( 1,193)( 2,196)( 3,195)( 4,194)( 5,201)( 6,204)( 7,203)( 8,202)
( 9,197)( 10,200)( 11,199)( 12,198)( 13,209)( 14,212)( 15,211)( 16,210)
( 17,205)( 18,208)( 19,207)( 20,206)( 21,213)( 22,216)( 23,215)( 24,214)
( 25,189)( 26,192)( 27,191)( 28,190)( 29,185)( 30,188)( 31,187)( 32,186)
( 33,181)( 34,184)( 35,183)( 36,182)( 37,153)( 38,156)( 39,155)( 40,154)
( 41,149)( 42,152)( 43,151)( 44,150)( 45,145)( 46,148)( 47,147)( 48,146)
( 49,157)( 50,160)( 51,159)( 52,158)( 53,165)( 54,168)( 55,167)( 56,166)
( 57,161)( 58,164)( 59,163)( 60,162)( 61,173)( 62,176)( 63,175)( 64,174)
( 65,169)( 66,172)( 67,171)( 68,170)( 69,177)( 70,180)( 71,179)( 72,178)
( 73,141)( 74,144)( 75,143)( 76,142)( 77,137)( 78,140)( 79,139)( 80,138)
( 81,133)( 82,136)( 83,135)( 84,134)( 85,109)( 86,112)( 87,111)( 88,110)
( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)( 94,116)( 95,115)( 96,114)
( 97,125)( 98,128)( 99,127)(100,126)(101,121)(102,124)(103,123)(104,122)
(105,129)(106,132)(107,131)(108,130)(217,305)(218,308)(219,307)(220,306)
(221,301)(222,304)(223,303)(224,302)(225,309)(226,312)(227,311)(228,310)
(229,321)(230,324)(231,323)(232,322)(233,317)(234,320)(235,319)(236,318)
(237,313)(238,316)(239,315)(240,314)(241,289)(242,292)(243,291)(244,290)
(245,297)(246,300)(247,299)(248,298)(249,293)(250,296)(251,295)(252,294)
(254,256)(257,261)(258,264)(259,263)(260,262)(265,269)(266,272)(267,271)
(268,270)(274,276)(277,285)(278,288)(279,287)(280,286)(282,284);;
s3 := ( 13, 29)( 14, 30)( 15, 31)( 16, 32)( 17, 33)( 18, 34)( 19, 35)( 20, 36)
( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 37, 73)( 38, 74)( 39, 75)( 40, 76)
( 41, 77)( 42, 78)( 43, 79)( 44, 80)( 45, 81)( 46, 82)( 47, 83)( 48, 84)
( 49,101)( 50,102)( 51,103)( 52,104)( 53,105)( 54,106)( 55,107)( 56,108)
( 57, 97)( 58, 98)( 59, 99)( 60,100)( 61, 93)( 62, 94)( 63, 95)( 64, 96)
( 65, 85)( 66, 86)( 67, 87)( 68, 88)( 69, 89)( 70, 90)( 71, 91)( 72, 92)
(121,137)(122,138)(123,139)(124,140)(125,141)(126,142)(127,143)(128,144)
(129,133)(130,134)(131,135)(132,136)(145,181)(146,182)(147,183)(148,184)
(149,185)(150,186)(151,187)(152,188)(153,189)(154,190)(155,191)(156,192)
(157,209)(158,210)(159,211)(160,212)(161,213)(162,214)(163,215)(164,216)
(165,205)(166,206)(167,207)(168,208)(169,201)(170,202)(171,203)(172,204)
(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)(180,200)
(229,245)(230,246)(231,247)(232,248)(233,249)(234,250)(235,251)(236,252)
(237,241)(238,242)(239,243)(240,244)(253,289)(254,290)(255,291)(256,292)
(257,293)(258,294)(259,295)(260,296)(261,297)(262,298)(263,299)(264,300)
(265,317)(266,318)(267,319)(268,320)(269,321)(270,322)(271,323)(272,324)
(273,313)(274,314)(275,315)(276,316)(277,309)(278,310)(279,311)(280,312)
(281,301)(282,302)(283,303)(284,304)(285,305)(286,306)(287,307)(288,308);;
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, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s2*s1*s3*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*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)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320)(321,323)(322,324);
s1 := 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);
s2 := Sym(324)!( 1,193)( 2,196)( 3,195)( 4,194)( 5,201)( 6,204)( 7,203)
( 8,202)( 9,197)( 10,200)( 11,199)( 12,198)( 13,209)( 14,212)( 15,211)
( 16,210)( 17,205)( 18,208)( 19,207)( 20,206)( 21,213)( 22,216)( 23,215)
( 24,214)( 25,189)( 26,192)( 27,191)( 28,190)( 29,185)( 30,188)( 31,187)
( 32,186)( 33,181)( 34,184)( 35,183)( 36,182)( 37,153)( 38,156)( 39,155)
( 40,154)( 41,149)( 42,152)( 43,151)( 44,150)( 45,145)( 46,148)( 47,147)
( 48,146)( 49,157)( 50,160)( 51,159)( 52,158)( 53,165)( 54,168)( 55,167)
( 56,166)( 57,161)( 58,164)( 59,163)( 60,162)( 61,173)( 62,176)( 63,175)
( 64,174)( 65,169)( 66,172)( 67,171)( 68,170)( 69,177)( 70,180)( 71,179)
( 72,178)( 73,141)( 74,144)( 75,143)( 76,142)( 77,137)( 78,140)( 79,139)
( 80,138)( 81,133)( 82,136)( 83,135)( 84,134)( 85,109)( 86,112)( 87,111)
( 88,110)( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)( 94,116)( 95,115)
( 96,114)( 97,125)( 98,128)( 99,127)(100,126)(101,121)(102,124)(103,123)
(104,122)(105,129)(106,132)(107,131)(108,130)(217,305)(218,308)(219,307)
(220,306)(221,301)(222,304)(223,303)(224,302)(225,309)(226,312)(227,311)
(228,310)(229,321)(230,324)(231,323)(232,322)(233,317)(234,320)(235,319)
(236,318)(237,313)(238,316)(239,315)(240,314)(241,289)(242,292)(243,291)
(244,290)(245,297)(246,300)(247,299)(248,298)(249,293)(250,296)(251,295)
(252,294)(254,256)(257,261)(258,264)(259,263)(260,262)(265,269)(266,272)
(267,271)(268,270)(274,276)(277,285)(278,288)(279,287)(280,286)(282,284);
s3 := Sym(324)!( 13, 29)( 14, 30)( 15, 31)( 16, 32)( 17, 33)( 18, 34)( 19, 35)
( 20, 36)( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 37, 73)( 38, 74)( 39, 75)
( 40, 76)( 41, 77)( 42, 78)( 43, 79)( 44, 80)( 45, 81)( 46, 82)( 47, 83)
( 48, 84)( 49,101)( 50,102)( 51,103)( 52,104)( 53,105)( 54,106)( 55,107)
( 56,108)( 57, 97)( 58, 98)( 59, 99)( 60,100)( 61, 93)( 62, 94)( 63, 95)
( 64, 96)( 65, 85)( 66, 86)( 67, 87)( 68, 88)( 69, 89)( 70, 90)( 71, 91)
( 72, 92)(121,137)(122,138)(123,139)(124,140)(125,141)(126,142)(127,143)
(128,144)(129,133)(130,134)(131,135)(132,136)(145,181)(146,182)(147,183)
(148,184)(149,185)(150,186)(151,187)(152,188)(153,189)(154,190)(155,191)
(156,192)(157,209)(158,210)(159,211)(160,212)(161,213)(162,214)(163,215)
(164,216)(165,205)(166,206)(167,207)(168,208)(169,201)(170,202)(171,203)
(172,204)(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)
(180,200)(229,245)(230,246)(231,247)(232,248)(233,249)(234,250)(235,251)
(236,252)(237,241)(238,242)(239,243)(240,244)(253,289)(254,290)(255,291)
(256,292)(257,293)(258,294)(259,295)(260,296)(261,297)(262,298)(263,299)
(264,300)(265,317)(266,318)(267,319)(268,320)(269,321)(270,322)(271,323)
(272,324)(273,313)(274,314)(275,315)(276,316)(277,309)(278,310)(279,311)
(280,312)(281,301)(282,302)(283,303)(284,304)(285,305)(286,306)(287,307)
(288,308);
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,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s2*s1*s3*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*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