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