Polytope of Type {6,6,6}

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