Polytope of Type {4,6,27}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,27}*1296
if this polytope has a name.
Group : SmallGroup(1296,904)
Rank : 4
Schlafli Type : {4,6,27}
Number of vertices, edges, etc : 4, 12, 81, 27
Order of s0s1s2s3 : 108
Order of s0s1s2s3s2s1 : 2
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 : {2,6,27}*648
3-fold quotients : {4,2,27}*432, {4,6,9}*432
6-fold quotients : {2,2,27}*216, {2,6,9}*216
9-fold quotients : {4,2,9}*144, {4,6,3}*144
18-fold quotients : {2,2,9}*72, {2,6,3}*72
27-fold quotients : {4,2,3}*48
54-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,163)( 2,164)( 3,165)( 4,166)( 5,167)( 6,168)( 7,169)( 8,170)
( 9,171)( 10,172)( 11,173)( 12,174)( 13,175)( 14,176)( 15,177)( 16,178)
( 17,179)( 18,180)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)( 24,186)
( 25,187)( 26,188)( 27,189)( 28,190)( 29,191)( 30,192)( 31,193)( 32,194)
( 33,195)( 34,196)( 35,197)( 36,198)( 37,199)( 38,200)( 39,201)( 40,202)
( 41,203)( 42,204)( 43,205)( 44,206)( 45,207)( 46,208)( 47,209)( 48,210)
( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,217)( 56,218)
( 57,219)( 58,220)( 59,221)( 60,222)( 61,223)( 62,224)( 63,225)( 64,226)
( 65,227)( 66,228)( 67,229)( 68,230)( 69,231)( 70,232)( 71,233)( 72,234)
( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)( 80,242)
( 81,243)( 82,244)( 83,245)( 84,246)( 85,247)( 86,248)( 87,249)( 88,250)
( 89,251)( 90,252)( 91,253)( 92,254)( 93,255)( 94,256)( 95,257)( 96,258)
( 97,259)( 98,260)( 99,261)(100,262)(101,263)(102,264)(103,265)(104,266)
(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)(112,274)
(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)(120,282)
(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)(128,290)
(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,298)
(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)(144,306)
(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)(152,314)
(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)
(161,323)(162,324);;
s1 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)(121,130)
(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)(147,156)
(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(163,244)(164,245)
(165,246)(166,247)(167,248)(168,249)(169,250)(170,251)(171,252)(172,262)
(173,263)(174,264)(175,265)(176,266)(177,267)(178,268)(179,269)(180,270)
(181,253)(182,254)(183,255)(184,256)(185,257)(186,258)(187,259)(188,260)
(189,261)(190,271)(191,272)(192,273)(193,274)(194,275)(195,276)(196,277)
(197,278)(198,279)(199,289)(200,290)(201,291)(202,292)(203,293)(204,294)
(205,295)(206,296)(207,297)(208,280)(209,281)(210,282)(211,283)(212,284)
(213,285)(214,286)(215,287)(216,288)(217,298)(218,299)(219,300)(220,301)
(221,302)(222,303)(223,304)(224,305)(225,306)(226,316)(227,317)(228,318)
(229,319)(230,320)(231,321)(232,322)(233,323)(234,324)(235,307)(236,308)
(237,309)(238,310)(239,311)(240,312)(241,313)(242,314)(243,315);;
s2 := ( 1, 10)( 2, 12)( 3, 11)( 4, 18)( 5, 17)( 6, 16)( 7, 15)( 8, 14)
( 9, 13)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 72)( 29, 71)( 30, 70)
( 31, 69)( 32, 68)( 33, 67)( 34, 66)( 35, 65)( 36, 64)( 37, 63)( 38, 62)
( 39, 61)( 40, 60)( 41, 59)( 42, 58)( 43, 57)( 44, 56)( 45, 55)( 46, 81)
( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)( 54, 73)
( 82, 91)( 83, 93)( 84, 92)( 85, 99)( 86, 98)( 87, 97)( 88, 96)( 89, 95)
( 90, 94)(101,102)(103,108)(104,107)(105,106)(109,153)(110,152)(111,151)
(112,150)(113,149)(114,148)(115,147)(116,146)(117,145)(118,144)(119,143)
(120,142)(121,141)(122,140)(123,139)(124,138)(125,137)(126,136)(127,162)
(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)
(163,172)(164,174)(165,173)(166,180)(167,179)(168,178)(169,177)(170,176)
(171,175)(182,183)(184,189)(185,188)(186,187)(190,234)(191,233)(192,232)
(193,231)(194,230)(195,229)(196,228)(197,227)(198,226)(199,225)(200,224)
(201,223)(202,222)(203,221)(204,220)(205,219)(206,218)(207,217)(208,243)
(209,242)(210,241)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)
(244,253)(245,255)(246,254)(247,261)(248,260)(249,259)(250,258)(251,257)
(252,256)(263,264)(265,270)(266,269)(267,268)(271,315)(272,314)(273,313)
(274,312)(275,311)(276,310)(277,309)(278,308)(279,307)(280,306)(281,305)
(282,304)(283,303)(284,302)(285,301)(286,300)(287,299)(288,298)(289,324)
(290,323)(291,322)(292,321)(293,320)(294,319)(295,318)(296,317)(297,316);;
s3 := ( 1, 28)( 2, 30)( 3, 29)( 4, 36)( 5, 35)( 6, 34)( 7, 33)( 8, 32)
( 9, 31)( 10, 46)( 11, 48)( 12, 47)( 13, 54)( 14, 53)( 15, 52)( 16, 51)
( 17, 50)( 18, 49)( 19, 37)( 20, 39)( 21, 38)( 22, 45)( 23, 44)( 24, 43)
( 25, 42)( 26, 41)( 27, 40)( 55, 63)( 56, 62)( 57, 61)( 58, 60)( 64, 81)
( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)
( 82,109)( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)( 89,113)
( 90,112)( 91,127)( 92,129)( 93,128)( 94,135)( 95,134)( 96,133)( 97,132)
( 98,131)( 99,130)(100,118)(101,120)(102,119)(103,126)(104,125)(105,124)
(106,123)(107,122)(108,121)(136,144)(137,143)(138,142)(139,141)(145,162)
(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)
(163,190)(164,192)(165,191)(166,198)(167,197)(168,196)(169,195)(170,194)
(171,193)(172,208)(173,210)(174,209)(175,216)(176,215)(177,214)(178,213)
(179,212)(180,211)(181,199)(182,201)(183,200)(184,207)(185,206)(186,205)
(187,204)(188,203)(189,202)(217,225)(218,224)(219,223)(220,222)(226,243)
(227,242)(228,241)(229,240)(230,239)(231,238)(232,237)(233,236)(234,235)
(244,271)(245,273)(246,272)(247,279)(248,278)(249,277)(250,276)(251,275)
(252,274)(253,289)(254,291)(255,290)(256,297)(257,296)(258,295)(259,294)
(260,293)(261,292)(262,280)(263,282)(264,281)(265,288)(266,287)(267,286)
(268,285)(269,284)(270,283)(298,306)(299,305)(300,304)(301,303)(307,324)
(308,323)(309,322)(310,321)(311,320)(312,319)(313,318)(314,317)(315,316);;
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*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(324)!( 1,163)( 2,164)( 3,165)( 4,166)( 5,167)( 6,168)( 7,169)
( 8,170)( 9,171)( 10,172)( 11,173)( 12,174)( 13,175)( 14,176)( 15,177)
( 16,178)( 17,179)( 18,180)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)
( 24,186)( 25,187)( 26,188)( 27,189)( 28,190)( 29,191)( 30,192)( 31,193)
( 32,194)( 33,195)( 34,196)( 35,197)( 36,198)( 37,199)( 38,200)( 39,201)
( 40,202)( 41,203)( 42,204)( 43,205)( 44,206)( 45,207)( 46,208)( 47,209)
( 48,210)( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,217)
( 56,218)( 57,219)( 58,220)( 59,221)( 60,222)( 61,223)( 62,224)( 63,225)
( 64,226)( 65,227)( 66,228)( 67,229)( 68,230)( 69,231)( 70,232)( 71,233)
( 72,234)( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)
( 80,242)( 81,243)( 82,244)( 83,245)( 84,246)( 85,247)( 86,248)( 87,249)
( 88,250)( 89,251)( 90,252)( 91,253)( 92,254)( 93,255)( 94,256)( 95,257)
( 96,258)( 97,259)( 98,260)( 99,261)(100,262)(101,263)(102,264)(103,265)
(104,266)(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)
(112,274)(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)
(120,282)(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)
(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)
(136,298)(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)
(144,306)(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)
(152,314)(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)
(160,322)(161,323)(162,324);
s1 := Sym(324)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)
(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)
(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(163,244)
(164,245)(165,246)(166,247)(167,248)(168,249)(169,250)(170,251)(171,252)
(172,262)(173,263)(174,264)(175,265)(176,266)(177,267)(178,268)(179,269)
(180,270)(181,253)(182,254)(183,255)(184,256)(185,257)(186,258)(187,259)
(188,260)(189,261)(190,271)(191,272)(192,273)(193,274)(194,275)(195,276)
(196,277)(197,278)(198,279)(199,289)(200,290)(201,291)(202,292)(203,293)
(204,294)(205,295)(206,296)(207,297)(208,280)(209,281)(210,282)(211,283)
(212,284)(213,285)(214,286)(215,287)(216,288)(217,298)(218,299)(219,300)
(220,301)(221,302)(222,303)(223,304)(224,305)(225,306)(226,316)(227,317)
(228,318)(229,319)(230,320)(231,321)(232,322)(233,323)(234,324)(235,307)
(236,308)(237,309)(238,310)(239,311)(240,312)(241,313)(242,314)(243,315);
s2 := Sym(324)!( 1, 10)( 2, 12)( 3, 11)( 4, 18)( 5, 17)( 6, 16)( 7, 15)
( 8, 14)( 9, 13)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 72)( 29, 71)
( 30, 70)( 31, 69)( 32, 68)( 33, 67)( 34, 66)( 35, 65)( 36, 64)( 37, 63)
( 38, 62)( 39, 61)( 40, 60)( 41, 59)( 42, 58)( 43, 57)( 44, 56)( 45, 55)
( 46, 81)( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)
( 54, 73)( 82, 91)( 83, 93)( 84, 92)( 85, 99)( 86, 98)( 87, 97)( 88, 96)
( 89, 95)( 90, 94)(101,102)(103,108)(104,107)(105,106)(109,153)(110,152)
(111,151)(112,150)(113,149)(114,148)(115,147)(116,146)(117,145)(118,144)
(119,143)(120,142)(121,141)(122,140)(123,139)(124,138)(125,137)(126,136)
(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)
(135,154)(163,172)(164,174)(165,173)(166,180)(167,179)(168,178)(169,177)
(170,176)(171,175)(182,183)(184,189)(185,188)(186,187)(190,234)(191,233)
(192,232)(193,231)(194,230)(195,229)(196,228)(197,227)(198,226)(199,225)
(200,224)(201,223)(202,222)(203,221)(204,220)(205,219)(206,218)(207,217)
(208,243)(209,242)(210,241)(211,240)(212,239)(213,238)(214,237)(215,236)
(216,235)(244,253)(245,255)(246,254)(247,261)(248,260)(249,259)(250,258)
(251,257)(252,256)(263,264)(265,270)(266,269)(267,268)(271,315)(272,314)
(273,313)(274,312)(275,311)(276,310)(277,309)(278,308)(279,307)(280,306)
(281,305)(282,304)(283,303)(284,302)(285,301)(286,300)(287,299)(288,298)
(289,324)(290,323)(291,322)(292,321)(293,320)(294,319)(295,318)(296,317)
(297,316);
s3 := Sym(324)!( 1, 28)( 2, 30)( 3, 29)( 4, 36)( 5, 35)( 6, 34)( 7, 33)
( 8, 32)( 9, 31)( 10, 46)( 11, 48)( 12, 47)( 13, 54)( 14, 53)( 15, 52)
( 16, 51)( 17, 50)( 18, 49)( 19, 37)( 20, 39)( 21, 38)( 22, 45)( 23, 44)
( 24, 43)( 25, 42)( 26, 41)( 27, 40)( 55, 63)( 56, 62)( 57, 61)( 58, 60)
( 64, 81)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)
( 72, 73)( 82,109)( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)
( 89,113)( 90,112)( 91,127)( 92,129)( 93,128)( 94,135)( 95,134)( 96,133)
( 97,132)( 98,131)( 99,130)(100,118)(101,120)(102,119)(103,126)(104,125)
(105,124)(106,123)(107,122)(108,121)(136,144)(137,143)(138,142)(139,141)
(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)
(153,154)(163,190)(164,192)(165,191)(166,198)(167,197)(168,196)(169,195)
(170,194)(171,193)(172,208)(173,210)(174,209)(175,216)(176,215)(177,214)
(178,213)(179,212)(180,211)(181,199)(182,201)(183,200)(184,207)(185,206)
(186,205)(187,204)(188,203)(189,202)(217,225)(218,224)(219,223)(220,222)
(226,243)(227,242)(228,241)(229,240)(230,239)(231,238)(232,237)(233,236)
(234,235)(244,271)(245,273)(246,272)(247,279)(248,278)(249,277)(250,276)
(251,275)(252,274)(253,289)(254,291)(255,290)(256,297)(257,296)(258,295)
(259,294)(260,293)(261,292)(262,280)(263,282)(264,281)(265,288)(266,287)
(267,286)(268,285)(269,284)(270,283)(298,306)(299,305)(300,304)(301,303)
(307,324)(308,323)(309,322)(310,321)(311,320)(312,319)(313,318)(314,317)
(315,316);
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*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope