Polytope of Type {4,27,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,27,6}*1296
if this polytope has a name.
Group : SmallGroup(1296,1781)
Rank : 4
Schlafli Type : {4,27,6}
Number of vertices, edges, etc : 4, 54, 81, 6
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 : {4,27,2}*432, {4,9,6}*432
   9-fold quotients : {4,9,2}*144, {4,3,6}*144
   27-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320)
(321,323)(322,324);;
s1 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 33)( 14, 34)( 15, 36)
( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)
( 24, 27)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 81)( 42, 82)( 43, 84)
( 44, 83)( 45, 77)( 46, 78)( 47, 80)( 48, 79)( 49,105)( 50,106)( 51,108)
( 52,107)( 53,101)( 54,102)( 55,104)( 56,103)( 57, 97)( 58, 98)( 59,100)
( 60, 99)( 61, 93)( 62, 94)( 63, 96)( 64, 95)( 65, 89)( 66, 90)( 67, 92)
( 68, 91)( 69, 85)( 70, 86)( 71, 88)( 72, 87)(109,249)(110,250)(111,252)
(112,251)(113,245)(114,246)(115,248)(116,247)(117,241)(118,242)(119,244)
(120,243)(121,237)(122,238)(123,240)(124,239)(125,233)(126,234)(127,236)
(128,235)(129,229)(130,230)(131,232)(132,231)(133,225)(134,226)(135,228)
(136,227)(137,221)(138,222)(139,224)(140,223)(141,217)(142,218)(143,220)
(144,219)(145,321)(146,322)(147,324)(148,323)(149,317)(150,318)(151,320)
(152,319)(153,313)(154,314)(155,316)(156,315)(157,309)(158,310)(159,312)
(160,311)(161,305)(162,306)(163,308)(164,307)(165,301)(166,302)(167,304)
(168,303)(169,297)(170,298)(171,300)(172,299)(173,293)(174,294)(175,296)
(176,295)(177,289)(178,290)(179,292)(180,291)(181,285)(182,286)(183,288)
(184,287)(185,281)(186,282)(187,284)(188,283)(189,277)(190,278)(191,280)
(192,279)(193,273)(194,274)(195,276)(196,275)(197,269)(198,270)(199,272)
(200,271)(201,265)(202,266)(203,268)(204,267)(205,261)(206,262)(207,264)
(208,263)(209,257)(210,258)(211,260)(212,259)(213,253)(214,254)(215,256)
(216,255);;
s2 := (  1,145)(  2,148)(  3,147)(  4,146)(  5,153)(  6,156)(  7,155)(  8,154)
(  9,149)( 10,152)( 11,151)( 12,150)( 13,177)( 14,180)( 15,179)( 16,178)
( 17,173)( 18,176)( 19,175)( 20,174)( 21,169)( 22,172)( 23,171)( 24,170)
( 25,165)( 26,168)( 27,167)( 28,166)( 29,161)( 30,164)( 31,163)( 32,162)
( 33,157)( 34,160)( 35,159)( 36,158)( 37,109)( 38,112)( 39,111)( 40,110)
( 41,117)( 42,120)( 43,119)( 44,118)( 45,113)( 46,116)( 47,115)( 48,114)
( 49,141)( 50,144)( 51,143)( 52,142)( 53,137)( 54,140)( 55,139)( 56,138)
( 57,133)( 58,136)( 59,135)( 60,134)( 61,129)( 62,132)( 63,131)( 64,130)
( 65,125)( 66,128)( 67,127)( 68,126)( 69,121)( 70,124)( 71,123)( 72,122)
( 73,181)( 74,184)( 75,183)( 76,182)( 77,189)( 78,192)( 79,191)( 80,190)
( 81,185)( 82,188)( 83,187)( 84,186)( 85,213)( 86,216)( 87,215)( 88,214)
( 89,209)( 90,212)( 91,211)( 92,210)( 93,205)( 94,208)( 95,207)( 96,206)
( 97,201)( 98,204)( 99,203)(100,202)(101,197)(102,200)(103,199)(104,198)
(105,193)(106,196)(107,195)(108,194)(217,285)(218,288)(219,287)(220,286)
(221,281)(222,284)(223,283)(224,282)(225,277)(226,280)(227,279)(228,278)
(229,273)(230,276)(231,275)(232,274)(233,269)(234,272)(235,271)(236,270)
(237,265)(238,268)(239,267)(240,266)(241,261)(242,264)(243,263)(244,262)
(245,257)(246,260)(247,259)(248,258)(249,253)(250,256)(251,255)(252,254)
(289,321)(290,324)(291,323)(292,322)(293,317)(294,320)(295,319)(296,318)
(297,313)(298,316)(299,315)(300,314)(301,309)(302,312)(303,311)(304,310)
(306,308);;
s3 := ( 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);;
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*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*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*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(324)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320)(321,323)(322,324);
s1 := Sym(324)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 33)( 14, 34)
( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)
( 23, 28)( 24, 27)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 81)( 42, 82)
( 43, 84)( 44, 83)( 45, 77)( 46, 78)( 47, 80)( 48, 79)( 49,105)( 50,106)
( 51,108)( 52,107)( 53,101)( 54,102)( 55,104)( 56,103)( 57, 97)( 58, 98)
( 59,100)( 60, 99)( 61, 93)( 62, 94)( 63, 96)( 64, 95)( 65, 89)( 66, 90)
( 67, 92)( 68, 91)( 69, 85)( 70, 86)( 71, 88)( 72, 87)(109,249)(110,250)
(111,252)(112,251)(113,245)(114,246)(115,248)(116,247)(117,241)(118,242)
(119,244)(120,243)(121,237)(122,238)(123,240)(124,239)(125,233)(126,234)
(127,236)(128,235)(129,229)(130,230)(131,232)(132,231)(133,225)(134,226)
(135,228)(136,227)(137,221)(138,222)(139,224)(140,223)(141,217)(142,218)
(143,220)(144,219)(145,321)(146,322)(147,324)(148,323)(149,317)(150,318)
(151,320)(152,319)(153,313)(154,314)(155,316)(156,315)(157,309)(158,310)
(159,312)(160,311)(161,305)(162,306)(163,308)(164,307)(165,301)(166,302)
(167,304)(168,303)(169,297)(170,298)(171,300)(172,299)(173,293)(174,294)
(175,296)(176,295)(177,289)(178,290)(179,292)(180,291)(181,285)(182,286)
(183,288)(184,287)(185,281)(186,282)(187,284)(188,283)(189,277)(190,278)
(191,280)(192,279)(193,273)(194,274)(195,276)(196,275)(197,269)(198,270)
(199,272)(200,271)(201,265)(202,266)(203,268)(204,267)(205,261)(206,262)
(207,264)(208,263)(209,257)(210,258)(211,260)(212,259)(213,253)(214,254)
(215,256)(216,255);
s2 := Sym(324)!(  1,145)(  2,148)(  3,147)(  4,146)(  5,153)(  6,156)(  7,155)
(  8,154)(  9,149)( 10,152)( 11,151)( 12,150)( 13,177)( 14,180)( 15,179)
( 16,178)( 17,173)( 18,176)( 19,175)( 20,174)( 21,169)( 22,172)( 23,171)
( 24,170)( 25,165)( 26,168)( 27,167)( 28,166)( 29,161)( 30,164)( 31,163)
( 32,162)( 33,157)( 34,160)( 35,159)( 36,158)( 37,109)( 38,112)( 39,111)
( 40,110)( 41,117)( 42,120)( 43,119)( 44,118)( 45,113)( 46,116)( 47,115)
( 48,114)( 49,141)( 50,144)( 51,143)( 52,142)( 53,137)( 54,140)( 55,139)
( 56,138)( 57,133)( 58,136)( 59,135)( 60,134)( 61,129)( 62,132)( 63,131)
( 64,130)( 65,125)( 66,128)( 67,127)( 68,126)( 69,121)( 70,124)( 71,123)
( 72,122)( 73,181)( 74,184)( 75,183)( 76,182)( 77,189)( 78,192)( 79,191)
( 80,190)( 81,185)( 82,188)( 83,187)( 84,186)( 85,213)( 86,216)( 87,215)
( 88,214)( 89,209)( 90,212)( 91,211)( 92,210)( 93,205)( 94,208)( 95,207)
( 96,206)( 97,201)( 98,204)( 99,203)(100,202)(101,197)(102,200)(103,199)
(104,198)(105,193)(106,196)(107,195)(108,194)(217,285)(218,288)(219,287)
(220,286)(221,281)(222,284)(223,283)(224,282)(225,277)(226,280)(227,279)
(228,278)(229,273)(230,276)(231,275)(232,274)(233,269)(234,272)(235,271)
(236,270)(237,265)(238,268)(239,267)(240,266)(241,261)(242,264)(243,263)
(244,262)(245,257)(246,260)(247,259)(248,258)(249,253)(250,256)(251,255)
(252,254)(289,321)(290,324)(291,323)(292,322)(293,317)(294,320)(295,319)
(296,318)(297,313)(298,316)(299,315)(300,314)(301,309)(302,312)(303,311)
(304,310)(306,308);
s3 := 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);
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*s0*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s3*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*s1*s2*s1*s2 >;

References : None.
to this polytope