Polytope of Type {6,3,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3,4}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1786)
Rank : 4
Schlafli Type : {6,3,4}
Number of vertices, edges, etc : 54, 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 : {6,3,4}*432
9-fold quotients : {6,3,4}*144
27-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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);;
s1 := ( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 45)( 6, 46)( 7, 48)( 8, 47)
( 9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 49)( 14, 50)( 15, 52)( 16, 51)
( 17, 57)( 18, 58)( 19, 60)( 20, 59)( 21, 53)( 22, 54)( 23, 56)( 24, 55)
( 25, 61)( 26, 62)( 27, 64)( 28, 63)( 29, 69)( 30, 70)( 31, 72)( 32, 71)
( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 75, 76)( 77, 81)( 78, 82)( 79, 84)
( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)( 99,100)(101,105)
(102,106)(103,108)(104,107)(109,261)(110,262)(111,264)(112,263)(113,257)
(114,258)(115,260)(116,259)(117,253)(118,254)(119,256)(120,255)(121,273)
(122,274)(123,276)(124,275)(125,269)(126,270)(127,272)(128,271)(129,265)
(130,266)(131,268)(132,267)(133,285)(134,286)(135,288)(136,287)(137,281)
(138,282)(139,284)(140,283)(141,277)(142,278)(143,280)(144,279)(145,225)
(146,226)(147,228)(148,227)(149,221)(150,222)(151,224)(152,223)(153,217)
(154,218)(155,220)(156,219)(157,237)(158,238)(159,240)(160,239)(161,233)
(162,234)(163,236)(164,235)(165,229)(166,230)(167,232)(168,231)(169,249)
(170,250)(171,252)(172,251)(173,245)(174,246)(175,248)(176,247)(177,241)
(178,242)(179,244)(180,243)(181,297)(182,298)(183,300)(184,299)(185,293)
(186,294)(187,296)(188,295)(189,289)(190,290)(191,292)(192,291)(193,309)
(194,310)(195,312)(196,311)(197,305)(198,306)(199,308)(200,307)(201,301)
(202,302)(203,304)(204,303)(205,321)(206,322)(207,324)(208,323)(209,317)
(210,318)(211,320)(212,319)(213,313)(214,314)(215,316)(216,315);;
s2 := ( 1,109)( 2,112)( 3,111)( 4,110)( 5,117)( 6,120)( 7,119)( 8,118)
( 9,113)( 10,116)( 11,115)( 12,114)( 13,129)( 14,132)( 15,131)( 16,130)
( 17,125)( 18,128)( 19,127)( 20,126)( 21,121)( 22,124)( 23,123)( 24,122)
( 25,137)( 26,140)( 27,139)( 28,138)( 29,133)( 30,136)( 31,135)( 32,134)
( 33,141)( 34,144)( 35,143)( 36,142)( 37,209)( 38,212)( 39,211)( 40,210)
( 41,205)( 42,208)( 43,207)( 44,206)( 45,213)( 46,216)( 47,215)( 48,214)
( 49,181)( 50,184)( 51,183)( 52,182)( 53,189)( 54,192)( 55,191)( 56,190)
( 57,185)( 58,188)( 59,187)( 60,186)( 61,201)( 62,204)( 63,203)( 64,202)
( 65,197)( 66,200)( 67,199)( 68,198)( 69,193)( 70,196)( 71,195)( 72,194)
( 73,157)( 74,160)( 75,159)( 76,158)( 77,165)( 78,168)( 79,167)( 80,166)
( 81,161)( 82,164)( 83,163)( 84,162)( 85,177)( 86,180)( 87,179)( 88,178)
( 89,173)( 90,176)( 91,175)( 92,174)( 93,169)( 94,172)( 95,171)( 96,170)
( 97,149)( 98,152)( 99,151)(100,150)(101,145)(102,148)(103,147)(104,146)
(105,153)(106,156)(107,155)(108,154)(217,225)(218,228)(219,227)(220,226)
(222,224)(229,233)(230,236)(231,235)(232,234)(238,240)(242,244)(245,249)
(246,252)(247,251)(248,250)(253,313)(254,316)(255,315)(256,314)(257,321)
(258,324)(259,323)(260,322)(261,317)(262,320)(263,319)(264,318)(265,297)
(266,300)(267,299)(268,298)(269,293)(270,296)(271,295)(272,294)(273,289)
(274,292)(275,291)(276,290)(277,305)(278,308)(279,307)(280,306)(281,301)
(282,304)(283,303)(284,302)(285,309)(286,312)(287,311)(288,310);;
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, s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s3*s2*s1*s3*s2*s1*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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);
s1 := Sym(324)!( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 45)( 6, 46)( 7, 48)
( 8, 47)( 9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 49)( 14, 50)( 15, 52)
( 16, 51)( 17, 57)( 18, 58)( 19, 60)( 20, 59)( 21, 53)( 22, 54)( 23, 56)
( 24, 55)( 25, 61)( 26, 62)( 27, 64)( 28, 63)( 29, 69)( 30, 70)( 31, 72)
( 32, 71)( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 75, 76)( 77, 81)( 78, 82)
( 79, 84)( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)( 99,100)
(101,105)(102,106)(103,108)(104,107)(109,261)(110,262)(111,264)(112,263)
(113,257)(114,258)(115,260)(116,259)(117,253)(118,254)(119,256)(120,255)
(121,273)(122,274)(123,276)(124,275)(125,269)(126,270)(127,272)(128,271)
(129,265)(130,266)(131,268)(132,267)(133,285)(134,286)(135,288)(136,287)
(137,281)(138,282)(139,284)(140,283)(141,277)(142,278)(143,280)(144,279)
(145,225)(146,226)(147,228)(148,227)(149,221)(150,222)(151,224)(152,223)
(153,217)(154,218)(155,220)(156,219)(157,237)(158,238)(159,240)(160,239)
(161,233)(162,234)(163,236)(164,235)(165,229)(166,230)(167,232)(168,231)
(169,249)(170,250)(171,252)(172,251)(173,245)(174,246)(175,248)(176,247)
(177,241)(178,242)(179,244)(180,243)(181,297)(182,298)(183,300)(184,299)
(185,293)(186,294)(187,296)(188,295)(189,289)(190,290)(191,292)(192,291)
(193,309)(194,310)(195,312)(196,311)(197,305)(198,306)(199,308)(200,307)
(201,301)(202,302)(203,304)(204,303)(205,321)(206,322)(207,324)(208,323)
(209,317)(210,318)(211,320)(212,319)(213,313)(214,314)(215,316)(216,315);
s2 := Sym(324)!( 1,109)( 2,112)( 3,111)( 4,110)( 5,117)( 6,120)( 7,119)
( 8,118)( 9,113)( 10,116)( 11,115)( 12,114)( 13,129)( 14,132)( 15,131)
( 16,130)( 17,125)( 18,128)( 19,127)( 20,126)( 21,121)( 22,124)( 23,123)
( 24,122)( 25,137)( 26,140)( 27,139)( 28,138)( 29,133)( 30,136)( 31,135)
( 32,134)( 33,141)( 34,144)( 35,143)( 36,142)( 37,209)( 38,212)( 39,211)
( 40,210)( 41,205)( 42,208)( 43,207)( 44,206)( 45,213)( 46,216)( 47,215)
( 48,214)( 49,181)( 50,184)( 51,183)( 52,182)( 53,189)( 54,192)( 55,191)
( 56,190)( 57,185)( 58,188)( 59,187)( 60,186)( 61,201)( 62,204)( 63,203)
( 64,202)( 65,197)( 66,200)( 67,199)( 68,198)( 69,193)( 70,196)( 71,195)
( 72,194)( 73,157)( 74,160)( 75,159)( 76,158)( 77,165)( 78,168)( 79,167)
( 80,166)( 81,161)( 82,164)( 83,163)( 84,162)( 85,177)( 86,180)( 87,179)
( 88,178)( 89,173)( 90,176)( 91,175)( 92,174)( 93,169)( 94,172)( 95,171)
( 96,170)( 97,149)( 98,152)( 99,151)(100,150)(101,145)(102,148)(103,147)
(104,146)(105,153)(106,156)(107,155)(108,154)(217,225)(218,228)(219,227)
(220,226)(222,224)(229,233)(230,236)(231,235)(232,234)(238,240)(242,244)
(245,249)(246,252)(247,251)(248,250)(253,313)(254,316)(255,315)(256,314)
(257,321)(258,324)(259,323)(260,322)(261,317)(262,320)(263,319)(264,318)
(265,297)(266,300)(267,299)(268,298)(269,293)(270,296)(271,295)(272,294)
(273,289)(274,292)(275,291)(276,290)(277,305)(278,308)(279,307)(280,306)
(281,301)(282,304)(283,303)(284,302)(285,309)(286,312)(287,311)(288,310);
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,
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s3*s2*s1*s3*s2*s1*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;
References : None.
to this polytope