Polytope of Type {4,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,9}*1296d
if this polytope has a name.
Group : SmallGroup(1296,930)
Rank : 4
Schlafli Type : {4,6,9}
Number of vertices, edges, etc : 4, 36, 81, 27
Order of s0s1s2s3 : 36
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 : {2,6,9}*648d
3-fold quotients : {4,6,3}*432a
6-fold quotients : {2,6,3}*216
9-fold quotients : {4,6,3}*144
18-fold quotients : {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.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s1*s2*s1*s2*s1*s3*s2*s1*s2*s3> of order 3.
9 facets:
9 of {4,6}*48a
4 vertex figures:
4 of 3-fold non-regular quotient of {6,9}*324d
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 := ( 4, 8)( 5, 9)( 6, 7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 85, 89)( 86, 90)( 87, 88)( 91,100)( 92,101)( 93,102)( 94,107)( 95,108)( 96,106)( 97,105)( 98,103)( 99,104)(112,116)(113,117)(114,115)(118,127)(119,128)(120,129)(121,134)(122,135)(123,133)(124,132)(125,130)(126,131)(139,143)(140,144)(141,142)(145,154)(146,155)(147,156)(148,161)(149,162)(150,160)(151,159)(152,157)(153,158)(163,244)(164,245)(165,246)(166,251)(167,252)(168,250)(169,249)(170,247)(171,248)(172,262)(173,263)(174,264)(175,269)(176,270)(177,268)(178,267)(179,265)(180,266)(181,253)(182,254)(183,255)(184,260)(185,261)(186,259)(187,258)(188,256)(189,257)(190,271)(191,272)(192,273)(193,278)(194,279)(195,277)(196,276)(197,274)(198,275)(199,289)(200,290)(201,291)(202,296)(203,297)(204,295)(205,294)(206,292)(207,293)(208,280)(209,281)(210,282)(211,287)(212,288)(213,286)(214,285)(215,283)(216,284)(217,298)(218,299)(219,300)(220,305)(221,306)(222,304)(223,303)(224,301)(225,302)(226,316)(227,317)(228,318)(229,323)(230,324)(231,322)(232,321)(233,319)(234,320)(235,307)(236,308)(237,309)(238,314)(239,315)(240,313)(241,312)(242,310)(243,311);;
s2 := ( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)( 8, 18)( 9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 65)( 29, 64)( 30, 66)( 31, 68)( 32, 67)( 33, 69)( 34, 71)( 35, 70)( 36, 72)( 37, 56)( 38, 55)( 39, 57)( 40, 59)( 41, 58)( 42, 60)( 43, 62)( 44, 61)( 45, 63)( 46, 74)( 47, 73)( 48, 75)( 49, 77)( 50, 76)( 51, 78)( 52, 80)( 53, 79)( 54, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)(107,108)(109,146)(110,145)(111,147)(112,149)(113,148)(114,150)(115,152)(116,151)(117,153)(118,137)(119,136)(120,138)(121,140)(122,139)(123,141)(124,143)(125,142)(126,144)(127,155)(128,154)(129,156)(130,158)(131,157)(132,159)(133,161)(134,160)(135,162)(163,172)(164,174)(165,173)(166,175)(167,177)(168,176)(169,178)(170,180)(171,179)(182,183)(185,186)(188,189)(190,227)(191,226)(192,228)(193,230)(194,229)(195,231)(196,233)(197,232)(198,234)(199,218)(200,217)(201,219)(202,221)(203,220)(204,222)(205,224)(206,223)(207,225)(208,236)(209,235)(210,237)(211,239)(212,238)(213,240)(214,242)(215,241)(216,243)(244,253)(245,255)(246,254)(247,256)(248,258)(249,257)(250,259)(251,261)(252,260)(263,264)(266,267)(269,270)(271,308)(272,307)(273,309)(274,311)(275,310)(276,312)(277,314)(278,313)(279,315)(280,299)(281,298)(282,300)(283,302)(284,301)(285,303)(286,305)(287,304)(288,306)(289,317)(290,316)(291,318)(292,320)(293,319)(294,321)(295,323)(296,322)(297,324);;
s3 := ( 1, 28)( 2, 30)( 3, 29)( 4, 33)( 5, 32)( 6, 31)( 7, 35)( 8, 34)( 9, 36)( 10, 53)( 11, 52)( 12, 54)( 13, 46)( 14, 48)( 15, 47)( 16, 51)( 17, 50)( 18, 49)( 19, 40)( 20, 42)( 21, 41)( 22, 45)( 23, 44)( 24, 43)( 25, 38)( 26, 37)( 27, 39)( 55, 56)( 59, 60)( 61, 63)( 64, 81)( 65, 80)( 66, 79)( 67, 74)( 68, 73)( 69, 75)( 70, 76)( 71, 78)( 72, 77)( 82,109)( 83,111)( 84,110)( 85,114)( 86,113)( 87,112)( 88,116)( 89,115)( 90,117)( 91,134)( 92,133)( 93,135)( 94,127)( 95,129)( 96,128)( 97,132)( 98,131)( 99,130)(100,121)(101,123)(102,122)(103,126)(104,125)(105,124)(106,119)(107,118)(108,120)(136,137)(140,141)(142,144)(145,162)(146,161)(147,160)(148,155)(149,154)(150,156)(151,157)(152,159)(153,158)(163,190)(164,192)(165,191)(166,195)(167,194)(168,193)(169,197)(170,196)(171,198)(172,215)(173,214)(174,216)(175,208)(176,210)(177,209)(178,213)(179,212)(180,211)(181,202)(182,204)(183,203)(184,207)(185,206)(186,205)(187,200)(188,199)(189,201)(217,218)(221,222)(223,225)(226,243)(227,242)(228,241)(229,236)(230,235)(231,237)(232,238)(233,240)(234,239)(244,271)(245,273)(246,272)(247,276)(248,275)(249,274)(250,278)(251,277)(252,279)(253,296)(254,295)(255,297)(256,289)(257,291)(258,290)(259,294)(260,293)(261,292)(262,283)(263,285)(264,284)(265,288)(266,287)(267,286)(268,281)(269,280)(270,282)(298,299)(302,303)(304,306)(307,324)(308,323)(309,322)(310,317)(311,316)(312,318)(313,319)(314,321)(315,320);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*s2*s1*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)!( 4, 8)( 5, 9)( 6, 7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 85, 89)( 86, 90)( 87, 88)( 91,100)( 92,101)( 93,102)( 94,107)( 95,108)( 96,106)( 97,105)( 98,103)( 99,104)(112,116)(113,117)(114,115)(118,127)(119,128)(120,129)(121,134)(122,135)(123,133)(124,132)(125,130)(126,131)(139,143)(140,144)(141,142)(145,154)(146,155)(147,156)(148,161)(149,162)(150,160)(151,159)(152,157)(153,158)(163,244)(164,245)(165,246)(166,251)(167,252)(168,250)(169,249)(170,247)(171,248)(172,262)(173,263)(174,264)(175,269)(176,270)(177,268)(178,267)(179,265)(180,266)(181,253)(182,254)(183,255)(184,260)(185,261)(186,259)(187,258)(188,256)(189,257)(190,271)(191,272)(192,273)(193,278)(194,279)(195,277)(196,276)(197,274)(198,275)(199,289)(200,290)(201,291)(202,296)(203,297)(204,295)(205,294)(206,292)(207,293)(208,280)(209,281)(210,282)(211,287)(212,288)(213,286)(214,285)(215,283)(216,284)(217,298)(218,299)(219,300)(220,305)(221,306)(222,304)(223,303)(224,301)(225,302)(226,316)(227,317)(228,318)(229,323)(230,324)(231,322)(232,321)(233,319)(234,320)(235,307)(236,308)(237,309)(238,314)(239,315)(240,313)(241,312)(242,310)(243,311);
s2 := Sym(324)!( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)( 8, 18)( 9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 65)( 29, 64)( 30, 66)( 31, 68)( 32, 67)( 33, 69)( 34, 71)( 35, 70)( 36, 72)( 37, 56)( 38, 55)( 39, 57)( 40, 59)( 41, 58)( 42, 60)( 43, 62)( 44, 61)( 45, 63)( 46, 74)( 47, 73)( 48, 75)( 49, 77)( 50, 76)( 51, 78)( 52, 80)( 53, 79)( 54, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)(107,108)(109,146)(110,145)(111,147)(112,149)(113,148)(114,150)(115,152)(116,151)(117,153)(118,137)(119,136)(120,138)(121,140)(122,139)(123,141)(124,143)(125,142)(126,144)(127,155)(128,154)(129,156)(130,158)(131,157)(132,159)(133,161)(134,160)(135,162)(163,172)(164,174)(165,173)(166,175)(167,177)(168,176)(169,178)(170,180)(171,179)(182,183)(185,186)(188,189)(190,227)(191,226)(192,228)(193,230)(194,229)(195,231)(196,233)(197,232)(198,234)(199,218)(200,217)(201,219)(202,221)(203,220)(204,222)(205,224)(206,223)(207,225)(208,236)(209,235)(210,237)(211,239)(212,238)(213,240)(214,242)(215,241)(216,243)(244,253)(245,255)(246,254)(247,256)(248,258)(249,257)(250,259)(251,261)(252,260)(263,264)(266,267)(269,270)(271,308)(272,307)(273,309)(274,311)(275,310)(276,312)(277,314)(278,313)(279,315)(280,299)(281,298)(282,300)(283,302)(284,301)(285,303)(286,305)(287,304)(288,306)(289,317)(290,316)(291,318)(292,320)(293,319)(294,321)(295,323)(296,322)(297,324);
s3 := Sym(324)!( 1, 28)( 2, 30)( 3, 29)( 4, 33)( 5, 32)( 6, 31)( 7, 35)( 8, 34)( 9, 36)( 10, 53)( 11, 52)( 12, 54)( 13, 46)( 14, 48)( 15, 47)( 16, 51)( 17, 50)( 18, 49)( 19, 40)( 20, 42)( 21, 41)( 22, 45)( 23, 44)( 24, 43)( 25, 38)( 26, 37)( 27, 39)( 55, 56)( 59, 60)( 61, 63)( 64, 81)( 65, 80)( 66, 79)( 67, 74)( 68, 73)( 69, 75)( 70, 76)( 71, 78)( 72, 77)( 82,109)( 83,111)( 84,110)( 85,114)( 86,113)( 87,112)( 88,116)( 89,115)( 90,117)( 91,134)( 92,133)( 93,135)( 94,127)( 95,129)( 96,128)( 97,132)( 98,131)( 99,130)(100,121)(101,123)(102,122)(103,126)(104,125)(105,124)(106,119)(107,118)(108,120)(136,137)(140,141)(142,144)(145,162)(146,161)(147,160)(148,155)(149,154)(150,156)(151,157)(152,159)(153,158)(163,190)(164,192)(165,191)(166,195)(167,194)(168,193)(169,197)(170,196)(171,198)(172,215)(173,214)(174,216)(175,208)(176,210)(177,209)(178,213)(179,212)(180,211)(181,202)(182,204)(183,203)(184,207)(185,206)(186,205)(187,200)(188,199)(189,201)(217,218)(221,222)(223,225)(226,243)(227,242)(228,241)(229,236)(230,235)(231,237)(232,238)(233,240)(234,239)(244,271)(245,273)(246,272)(247,276)(248,275)(249,274)(250,278)(251,277)(252,279)(253,296)(254,295)(255,297)(256,289)(257,291)(258,290)(259,294)(260,293)(261,292)(262,283)(263,285)(264,284)(265,288)(266,287)(267,286)(268,281)(269,280)(270,282)(298,299)(302,303)(304,306)(307,324)(308,323)(309,322)(310,317)(311,316)(312,318)(313,319)(314,321)(315,320);
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*s2*s1*s3*s2*s3 >;
References : None.
to this polytope