Polytope of Type {6,6,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*1296b
if this polytope has a name.
Group : SmallGroup(1296,1860)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 6, 54, 54, 18
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 6
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 : {3,6,6}*648b, {6,6,3}*648b
3-fold quotients : {6,6,6}*432a
4-fold quotients : {3,6,3}*324a
6-fold quotients : {3,6,6}*216a, {6,6,3}*216a
9-fold quotients : {6,2,6}*144
12-fold quotients : {3,6,3}*108
18-fold quotients : {3,2,6}*72, {6,2,3}*72
27-fold quotients : {2,2,6}*48, {6,2,2}*48
36-fold quotients : {3,2,3}*36
54-fold quotients : {2,2,3}*24, {3,2,2}*24
81-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)(195,224)
(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)(203,234)
(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)(211,241)
(212,243)(213,242)(214,238)(215,240)(216,239)(245,246)(247,250)(248,252)
(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)(266,270)
(267,269)(271,298)(272,300)(273,299)(274,304)(275,306)(276,305)(277,301)
(278,303)(279,302)(280,307)(281,309)(282,308)(283,313)(284,315)(285,314)
(286,310)(287,312)(288,311)(289,316)(290,318)(291,317)(292,322)(293,324)
(294,323)(295,319)(296,321)(297,320);;
s1 := ( 1,190)( 2,192)( 3,191)( 4,196)( 5,198)( 6,197)( 7,193)( 8,195)
( 9,194)( 10,203)( 11,202)( 12,204)( 13,200)( 14,199)( 15,201)( 16,206)
( 17,205)( 18,207)( 19,215)( 20,214)( 21,216)( 22,212)( 23,211)( 24,213)
( 25,209)( 26,208)( 27,210)( 28,163)( 29,165)( 30,164)( 31,169)( 32,171)
( 33,170)( 34,166)( 35,168)( 36,167)( 37,176)( 38,175)( 39,177)( 40,173)
( 41,172)( 42,174)( 43,179)( 44,178)( 45,180)( 46,188)( 47,187)( 48,189)
( 49,185)( 50,184)( 51,186)( 52,182)( 53,181)( 54,183)( 55,217)( 56,219)
( 57,218)( 58,223)( 59,225)( 60,224)( 61,220)( 62,222)( 63,221)( 64,230)
( 65,229)( 66,231)( 67,227)( 68,226)( 69,228)( 70,233)( 71,232)( 72,234)
( 73,242)( 74,241)( 75,243)( 76,239)( 77,238)( 78,240)( 79,236)( 80,235)
( 81,237)( 82,271)( 83,273)( 84,272)( 85,277)( 86,279)( 87,278)( 88,274)
( 89,276)( 90,275)( 91,284)( 92,283)( 93,285)( 94,281)( 95,280)( 96,282)
( 97,287)( 98,286)( 99,288)(100,296)(101,295)(102,297)(103,293)(104,292)
(105,294)(106,290)(107,289)(108,291)(109,244)(110,246)(111,245)(112,250)
(113,252)(114,251)(115,247)(116,249)(117,248)(118,257)(119,256)(120,258)
(121,254)(122,253)(123,255)(124,260)(125,259)(126,261)(127,269)(128,268)
(129,270)(130,266)(131,265)(132,267)(133,263)(134,262)(135,264)(136,298)
(137,300)(138,299)(139,304)(140,306)(141,305)(142,301)(143,303)(144,302)
(145,311)(146,310)(147,312)(148,308)(149,307)(150,309)(151,314)(152,313)
(153,315)(154,323)(155,322)(156,324)(157,320)(158,319)(159,321)(160,317)
(161,316)(162,318);;
s2 := ( 1, 10)( 2, 11)( 3, 12)( 4, 17)( 5, 18)( 6, 16)( 7, 15)( 8, 13)
( 9, 14)( 22, 26)( 23, 27)( 24, 25)( 28, 37)( 29, 38)( 30, 39)( 31, 44)
( 32, 45)( 33, 43)( 34, 42)( 35, 40)( 36, 41)( 49, 53)( 50, 54)( 51, 52)
( 55, 64)( 56, 65)( 57, 66)( 58, 71)( 59, 72)( 60, 70)( 61, 69)( 62, 67)
( 63, 68)( 76, 80)( 77, 81)( 78, 79)( 82, 91)( 83, 92)( 84, 93)( 85, 98)
( 86, 99)( 87, 97)( 88, 96)( 89, 94)( 90, 95)(103,107)(104,108)(105,106)
(109,118)(110,119)(111,120)(112,125)(113,126)(114,124)(115,123)(116,121)
(117,122)(130,134)(131,135)(132,133)(136,145)(137,146)(138,147)(139,152)
(140,153)(141,151)(142,150)(143,148)(144,149)(157,161)(158,162)(159,160)
(163,172)(164,173)(165,174)(166,179)(167,180)(168,178)(169,177)(170,175)
(171,176)(184,188)(185,189)(186,187)(190,199)(191,200)(192,201)(193,206)
(194,207)(195,205)(196,204)(197,202)(198,203)(211,215)(212,216)(213,214)
(217,226)(218,227)(219,228)(220,233)(221,234)(222,232)(223,231)(224,229)
(225,230)(238,242)(239,243)(240,241)(244,253)(245,254)(246,255)(247,260)
(248,261)(249,259)(250,258)(251,256)(252,257)(265,269)(266,270)(267,268)
(271,280)(272,281)(273,282)(274,287)(275,288)(276,286)(277,285)(278,283)
(279,284)(292,296)(293,297)(294,295)(298,307)(299,308)(300,309)(301,314)
(302,315)(303,313)(304,312)(305,310)(306,311)(319,323)(320,324)(321,322);;
s3 := ( 1, 82)( 2, 83)( 3, 84)( 4, 88)( 5, 89)( 6, 90)( 7, 85)( 8, 86)
( 9, 87)( 10,100)( 11,101)( 12,102)( 13,106)( 14,107)( 15,108)( 16,103)
( 17,104)( 18,105)( 19, 91)( 20, 92)( 21, 93)( 22, 97)( 23, 98)( 24, 99)
( 25, 94)( 26, 95)( 27, 96)( 28,109)( 29,110)( 30,111)( 31,115)( 32,116)
( 33,117)( 34,112)( 35,113)( 36,114)( 37,127)( 38,128)( 39,129)( 40,133)
( 41,134)( 42,135)( 43,130)( 44,131)( 45,132)( 46,118)( 47,119)( 48,120)
( 49,124)( 50,125)( 51,126)( 52,121)( 53,122)( 54,123)( 55,136)( 56,137)
( 57,138)( 58,142)( 59,143)( 60,144)( 61,139)( 62,140)( 63,141)( 64,154)
( 65,155)( 66,156)( 67,160)( 68,161)( 69,162)( 70,157)( 71,158)( 72,159)
( 73,145)( 74,146)( 75,147)( 76,151)( 77,152)( 78,153)( 79,148)( 80,149)
( 81,150)(163,244)(164,245)(165,246)(166,250)(167,251)(168,252)(169,247)
(170,248)(171,249)(172,262)(173,263)(174,264)(175,268)(176,269)(177,270)
(178,265)(179,266)(180,267)(181,253)(182,254)(183,255)(184,259)(185,260)
(186,261)(187,256)(188,257)(189,258)(190,271)(191,272)(192,273)(193,277)
(194,278)(195,279)(196,274)(197,275)(198,276)(199,289)(200,290)(201,291)
(202,295)(203,296)(204,297)(205,292)(206,293)(207,294)(208,280)(209,281)
(210,282)(211,286)(212,287)(213,288)(214,283)(215,284)(216,285)(217,298)
(218,299)(219,300)(220,304)(221,305)(222,306)(223,301)(224,302)(225,303)
(226,316)(227,317)(228,318)(229,322)(230,323)(231,324)(232,319)(233,320)
(234,321)(235,307)(236,308)(237,309)(238,313)(239,314)(240,315)(241,310)
(242,311)(243,312);;
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*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)
(195,224)(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)
(203,234)(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)
(211,241)(212,243)(213,242)(214,238)(215,240)(216,239)(245,246)(247,250)
(248,252)(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)
(266,270)(267,269)(271,298)(272,300)(273,299)(274,304)(275,306)(276,305)
(277,301)(278,303)(279,302)(280,307)(281,309)(282,308)(283,313)(284,315)
(285,314)(286,310)(287,312)(288,311)(289,316)(290,318)(291,317)(292,322)
(293,324)(294,323)(295,319)(296,321)(297,320);
s1 := Sym(324)!( 1,190)( 2,192)( 3,191)( 4,196)( 5,198)( 6,197)( 7,193)
( 8,195)( 9,194)( 10,203)( 11,202)( 12,204)( 13,200)( 14,199)( 15,201)
( 16,206)( 17,205)( 18,207)( 19,215)( 20,214)( 21,216)( 22,212)( 23,211)
( 24,213)( 25,209)( 26,208)( 27,210)( 28,163)( 29,165)( 30,164)( 31,169)
( 32,171)( 33,170)( 34,166)( 35,168)( 36,167)( 37,176)( 38,175)( 39,177)
( 40,173)( 41,172)( 42,174)( 43,179)( 44,178)( 45,180)( 46,188)( 47,187)
( 48,189)( 49,185)( 50,184)( 51,186)( 52,182)( 53,181)( 54,183)( 55,217)
( 56,219)( 57,218)( 58,223)( 59,225)( 60,224)( 61,220)( 62,222)( 63,221)
( 64,230)( 65,229)( 66,231)( 67,227)( 68,226)( 69,228)( 70,233)( 71,232)
( 72,234)( 73,242)( 74,241)( 75,243)( 76,239)( 77,238)( 78,240)( 79,236)
( 80,235)( 81,237)( 82,271)( 83,273)( 84,272)( 85,277)( 86,279)( 87,278)
( 88,274)( 89,276)( 90,275)( 91,284)( 92,283)( 93,285)( 94,281)( 95,280)
( 96,282)( 97,287)( 98,286)( 99,288)(100,296)(101,295)(102,297)(103,293)
(104,292)(105,294)(106,290)(107,289)(108,291)(109,244)(110,246)(111,245)
(112,250)(113,252)(114,251)(115,247)(116,249)(117,248)(118,257)(119,256)
(120,258)(121,254)(122,253)(123,255)(124,260)(125,259)(126,261)(127,269)
(128,268)(129,270)(130,266)(131,265)(132,267)(133,263)(134,262)(135,264)
(136,298)(137,300)(138,299)(139,304)(140,306)(141,305)(142,301)(143,303)
(144,302)(145,311)(146,310)(147,312)(148,308)(149,307)(150,309)(151,314)
(152,313)(153,315)(154,323)(155,322)(156,324)(157,320)(158,319)(159,321)
(160,317)(161,316)(162,318);
s2 := Sym(324)!( 1, 10)( 2, 11)( 3, 12)( 4, 17)( 5, 18)( 6, 16)( 7, 15)
( 8, 13)( 9, 14)( 22, 26)( 23, 27)( 24, 25)( 28, 37)( 29, 38)( 30, 39)
( 31, 44)( 32, 45)( 33, 43)( 34, 42)( 35, 40)( 36, 41)( 49, 53)( 50, 54)
( 51, 52)( 55, 64)( 56, 65)( 57, 66)( 58, 71)( 59, 72)( 60, 70)( 61, 69)
( 62, 67)( 63, 68)( 76, 80)( 77, 81)( 78, 79)( 82, 91)( 83, 92)( 84, 93)
( 85, 98)( 86, 99)( 87, 97)( 88, 96)( 89, 94)( 90, 95)(103,107)(104,108)
(105,106)(109,118)(110,119)(111,120)(112,125)(113,126)(114,124)(115,123)
(116,121)(117,122)(130,134)(131,135)(132,133)(136,145)(137,146)(138,147)
(139,152)(140,153)(141,151)(142,150)(143,148)(144,149)(157,161)(158,162)
(159,160)(163,172)(164,173)(165,174)(166,179)(167,180)(168,178)(169,177)
(170,175)(171,176)(184,188)(185,189)(186,187)(190,199)(191,200)(192,201)
(193,206)(194,207)(195,205)(196,204)(197,202)(198,203)(211,215)(212,216)
(213,214)(217,226)(218,227)(219,228)(220,233)(221,234)(222,232)(223,231)
(224,229)(225,230)(238,242)(239,243)(240,241)(244,253)(245,254)(246,255)
(247,260)(248,261)(249,259)(250,258)(251,256)(252,257)(265,269)(266,270)
(267,268)(271,280)(272,281)(273,282)(274,287)(275,288)(276,286)(277,285)
(278,283)(279,284)(292,296)(293,297)(294,295)(298,307)(299,308)(300,309)
(301,314)(302,315)(303,313)(304,312)(305,310)(306,311)(319,323)(320,324)
(321,322);
s3 := Sym(324)!( 1, 82)( 2, 83)( 3, 84)( 4, 88)( 5, 89)( 6, 90)( 7, 85)
( 8, 86)( 9, 87)( 10,100)( 11,101)( 12,102)( 13,106)( 14,107)( 15,108)
( 16,103)( 17,104)( 18,105)( 19, 91)( 20, 92)( 21, 93)( 22, 97)( 23, 98)
( 24, 99)( 25, 94)( 26, 95)( 27, 96)( 28,109)( 29,110)( 30,111)( 31,115)
( 32,116)( 33,117)( 34,112)( 35,113)( 36,114)( 37,127)( 38,128)( 39,129)
( 40,133)( 41,134)( 42,135)( 43,130)( 44,131)( 45,132)( 46,118)( 47,119)
( 48,120)( 49,124)( 50,125)( 51,126)( 52,121)( 53,122)( 54,123)( 55,136)
( 56,137)( 57,138)( 58,142)( 59,143)( 60,144)( 61,139)( 62,140)( 63,141)
( 64,154)( 65,155)( 66,156)( 67,160)( 68,161)( 69,162)( 70,157)( 71,158)
( 72,159)( 73,145)( 74,146)( 75,147)( 76,151)( 77,152)( 78,153)( 79,148)
( 80,149)( 81,150)(163,244)(164,245)(165,246)(166,250)(167,251)(168,252)
(169,247)(170,248)(171,249)(172,262)(173,263)(174,264)(175,268)(176,269)
(177,270)(178,265)(179,266)(180,267)(181,253)(182,254)(183,255)(184,259)
(185,260)(186,261)(187,256)(188,257)(189,258)(190,271)(191,272)(192,273)
(193,277)(194,278)(195,279)(196,274)(197,275)(198,276)(199,289)(200,290)
(201,291)(202,295)(203,296)(204,297)(205,292)(206,293)(207,294)(208,280)
(209,281)(210,282)(211,286)(212,287)(213,288)(214,283)(215,284)(216,285)
(217,298)(218,299)(219,300)(220,304)(221,305)(222,306)(223,301)(224,302)
(225,303)(226,316)(227,317)(228,318)(229,322)(230,323)(231,324)(232,319)
(233,320)(234,321)(235,307)(236,308)(237,309)(238,313)(239,314)(240,315)
(241,310)(242,311)(243,312);
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*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope