Polytope of Type {6,27,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,27,4}*1296
if this polytope has a name.
Group : SmallGroup(1296,1781)
Rank : 4
Schlafli Type : {6,27,4}
Number of vertices, edges, etc : 6, 81, 54, 4
Order of s0s1s2s3 : 54
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 : {2,27,4}*432, {6,9,4}*432
9-fold quotients : {2,9,4}*144, {6,3,4}*144
27-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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, 85)( 50, 86)( 51, 87)( 52, 88)
( 53, 89)( 54, 90)( 55, 91)( 56, 92)( 57, 93)( 58, 94)( 59, 95)( 60, 96)
( 61, 97)( 62, 98)( 63, 99)( 64,100)( 65,101)( 66,102)( 67,103)( 68,104)
( 69,105)( 70,106)( 71,107)( 72,108)(145,181)(146,182)(147,183)(148,184)
(149,185)(150,186)(151,187)(152,188)(153,189)(154,190)(155,191)(156,192)
(157,193)(158,194)(159,195)(160,196)(161,197)(162,198)(163,199)(164,200)
(165,201)(166,202)(167,203)(168,204)(169,205)(170,206)(171,207)(172,208)
(173,209)(174,210)(175,211)(176,212)(177,213)(178,214)(179,215)(180,216)
(253,289)(254,290)(255,291)(256,292)(257,293)(258,294)(259,295)(260,296)
(261,297)(262,298)(263,299)(264,300)(265,301)(266,302)(267,303)(268,304)
(269,305)(270,306)(271,307)(272,308)(273,309)(274,310)(275,311)(276,312)
(277,313)(278,314)(279,315)(280,316)(281,317)(282,318)(283,319)(284,320)
(285,321)(286,322)(287,323)(288,324);;
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, 69)( 14, 70)( 15, 72)( 16, 71)
( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 61)( 22, 62)( 23, 64)( 24, 63)
( 25, 57)( 26, 58)( 27, 60)( 28, 59)( 29, 53)( 30, 54)( 31, 56)( 32, 55)
( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 75, 76)( 77, 81)( 78, 82)( 79, 84)
( 80, 83)( 85,105)( 86,106)( 87,108)( 88,107)( 89,101)( 90,102)( 91,104)
( 92,103)( 93, 97)( 94, 98)( 95,100)( 96, 99)(109,285)(110,286)(111,288)
(112,287)(113,281)(114,282)(115,284)(116,283)(117,277)(118,278)(119,280)
(120,279)(121,273)(122,274)(123,276)(124,275)(125,269)(126,270)(127,272)
(128,271)(129,265)(130,266)(131,268)(132,267)(133,261)(134,262)(135,264)
(136,263)(137,257)(138,258)(139,260)(140,259)(141,253)(142,254)(143,256)
(144,255)(145,249)(146,250)(147,252)(148,251)(149,245)(150,246)(151,248)
(152,247)(153,241)(154,242)(155,244)(156,243)(157,237)(158,238)(159,240)
(160,239)(161,233)(162,234)(163,236)(164,235)(165,229)(166,230)(167,232)
(168,231)(169,225)(170,226)(171,228)(172,227)(173,221)(174,222)(175,224)
(176,223)(177,217)(178,218)(179,220)(180,219)(181,321)(182,322)(183,324)
(184,323)(185,317)(186,318)(187,320)(188,319)(189,313)(190,314)(191,316)
(192,315)(193,309)(194,310)(195,312)(196,311)(197,305)(198,306)(199,308)
(200,307)(201,301)(202,302)(203,304)(204,303)(205,297)(206,298)(207,300)
(208,299)(209,293)(210,294)(211,296)(212,295)(213,289)(214,290)(215,292)
(216,291);;
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)( 13,141)( 14,144)( 15,143)( 16,142)
( 17,137)( 18,140)( 19,139)( 20,138)( 21,133)( 22,136)( 23,135)( 24,134)
( 25,129)( 26,132)( 27,131)( 28,130)( 29,125)( 30,128)( 31,127)( 32,126)
( 33,121)( 34,124)( 35,123)( 36,122)( 37,181)( 38,184)( 39,183)( 40,182)
( 41,189)( 42,192)( 43,191)( 44,190)( 45,185)( 46,188)( 47,187)( 48,186)
( 49,213)( 50,216)( 51,215)( 52,214)( 53,209)( 54,212)( 55,211)( 56,210)
( 57,205)( 58,208)( 59,207)( 60,206)( 61,201)( 62,204)( 63,203)( 64,202)
( 65,197)( 66,200)( 67,199)( 68,198)( 69,193)( 70,196)( 71,195)( 72,194)
( 73,145)( 74,148)( 75,147)( 76,146)( 77,153)( 78,156)( 79,155)( 80,154)
( 81,149)( 82,152)( 83,151)( 84,150)( 85,177)( 86,180)( 87,179)( 88,178)
( 89,173)( 90,176)( 91,175)( 92,174)( 93,169)( 94,172)( 95,171)( 96,170)
( 97,165)( 98,168)( 99,167)(100,166)(101,161)(102,164)(103,163)(104,162)
(105,157)(106,160)(107,159)(108,158)(217,249)(218,252)(219,251)(220,250)
(221,245)(222,248)(223,247)(224,246)(225,241)(226,244)(227,243)(228,242)
(229,237)(230,240)(231,239)(232,238)(234,236)(253,321)(254,324)(255,323)
(256,322)(257,317)(258,320)(259,319)(260,318)(261,313)(262,316)(263,315)
(264,314)(265,309)(266,312)(267,311)(268,310)(269,305)(270,308)(271,307)
(272,306)(273,301)(274,304)(275,303)(276,302)(277,297)(278,300)(279,299)
(280,298)(281,293)(282,296)(283,295)(284,294)(285,289)(286,292)(287,291)
(288,290);;
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, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s2*s1*s3*s2*s3*s2*s1*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!( 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, 85)( 50, 86)( 51, 87)
( 52, 88)( 53, 89)( 54, 90)( 55, 91)( 56, 92)( 57, 93)( 58, 94)( 59, 95)
( 60, 96)( 61, 97)( 62, 98)( 63, 99)( 64,100)( 65,101)( 66,102)( 67,103)
( 68,104)( 69,105)( 70,106)( 71,107)( 72,108)(145,181)(146,182)(147,183)
(148,184)(149,185)(150,186)(151,187)(152,188)(153,189)(154,190)(155,191)
(156,192)(157,193)(158,194)(159,195)(160,196)(161,197)(162,198)(163,199)
(164,200)(165,201)(166,202)(167,203)(168,204)(169,205)(170,206)(171,207)
(172,208)(173,209)(174,210)(175,211)(176,212)(177,213)(178,214)(179,215)
(180,216)(253,289)(254,290)(255,291)(256,292)(257,293)(258,294)(259,295)
(260,296)(261,297)(262,298)(263,299)(264,300)(265,301)(266,302)(267,303)
(268,304)(269,305)(270,306)(271,307)(272,308)(273,309)(274,310)(275,311)
(276,312)(277,313)(278,314)(279,315)(280,316)(281,317)(282,318)(283,319)
(284,320)(285,321)(286,322)(287,323)(288,324);
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, 69)( 14, 70)( 15, 72)
( 16, 71)( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 61)( 22, 62)( 23, 64)
( 24, 63)( 25, 57)( 26, 58)( 27, 60)( 28, 59)( 29, 53)( 30, 54)( 31, 56)
( 32, 55)( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 75, 76)( 77, 81)( 78, 82)
( 79, 84)( 80, 83)( 85,105)( 86,106)( 87,108)( 88,107)( 89,101)( 90,102)
( 91,104)( 92,103)( 93, 97)( 94, 98)( 95,100)( 96, 99)(109,285)(110,286)
(111,288)(112,287)(113,281)(114,282)(115,284)(116,283)(117,277)(118,278)
(119,280)(120,279)(121,273)(122,274)(123,276)(124,275)(125,269)(126,270)
(127,272)(128,271)(129,265)(130,266)(131,268)(132,267)(133,261)(134,262)
(135,264)(136,263)(137,257)(138,258)(139,260)(140,259)(141,253)(142,254)
(143,256)(144,255)(145,249)(146,250)(147,252)(148,251)(149,245)(150,246)
(151,248)(152,247)(153,241)(154,242)(155,244)(156,243)(157,237)(158,238)
(159,240)(160,239)(161,233)(162,234)(163,236)(164,235)(165,229)(166,230)
(167,232)(168,231)(169,225)(170,226)(171,228)(172,227)(173,221)(174,222)
(175,224)(176,223)(177,217)(178,218)(179,220)(180,219)(181,321)(182,322)
(183,324)(184,323)(185,317)(186,318)(187,320)(188,319)(189,313)(190,314)
(191,316)(192,315)(193,309)(194,310)(195,312)(196,311)(197,305)(198,306)
(199,308)(200,307)(201,301)(202,302)(203,304)(204,303)(205,297)(206,298)
(207,300)(208,299)(209,293)(210,294)(211,296)(212,295)(213,289)(214,290)
(215,292)(216,291);
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)( 13,141)( 14,144)( 15,143)
( 16,142)( 17,137)( 18,140)( 19,139)( 20,138)( 21,133)( 22,136)( 23,135)
( 24,134)( 25,129)( 26,132)( 27,131)( 28,130)( 29,125)( 30,128)( 31,127)
( 32,126)( 33,121)( 34,124)( 35,123)( 36,122)( 37,181)( 38,184)( 39,183)
( 40,182)( 41,189)( 42,192)( 43,191)( 44,190)( 45,185)( 46,188)( 47,187)
( 48,186)( 49,213)( 50,216)( 51,215)( 52,214)( 53,209)( 54,212)( 55,211)
( 56,210)( 57,205)( 58,208)( 59,207)( 60,206)( 61,201)( 62,204)( 63,203)
( 64,202)( 65,197)( 66,200)( 67,199)( 68,198)( 69,193)( 70,196)( 71,195)
( 72,194)( 73,145)( 74,148)( 75,147)( 76,146)( 77,153)( 78,156)( 79,155)
( 80,154)( 81,149)( 82,152)( 83,151)( 84,150)( 85,177)( 86,180)( 87,179)
( 88,178)( 89,173)( 90,176)( 91,175)( 92,174)( 93,169)( 94,172)( 95,171)
( 96,170)( 97,165)( 98,168)( 99,167)(100,166)(101,161)(102,164)(103,163)
(104,162)(105,157)(106,160)(107,159)(108,158)(217,249)(218,252)(219,251)
(220,250)(221,245)(222,248)(223,247)(224,246)(225,241)(226,244)(227,243)
(228,242)(229,237)(230,240)(231,239)(232,238)(234,236)(253,321)(254,324)
(255,323)(256,322)(257,317)(258,320)(259,319)(260,318)(261,313)(262,316)
(263,315)(264,314)(265,309)(266,312)(267,311)(268,310)(269,305)(270,308)
(271,307)(272,306)(273,301)(274,304)(275,303)(276,302)(277,297)(278,300)
(279,299)(280,298)(281,293)(282,296)(283,295)(284,294)(285,289)(286,292)
(287,291)(288,290);
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,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope