Polytope of Type {4,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,9}*1296c
if this polytope has a name.
Group : SmallGroup(1296,917)
Rank : 4
Schlafli Type : {4,6,9}
Number of vertices, edges, etc : 4, 36, 81, 27
Order of s0s1s2s3 : 12
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}*648b
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}*324b
P/N, where N=<s1*s2*s1*s2> of order 3.
15 facets:
9 of {4,2}*16
6 of {4,6}*48a
4 vertex figures:
4 of 3-fold non-regular quotient of {6,9}*324b
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 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 55)( 29, 57)( 30, 56)( 31, 61)( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)( 47, 75)( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(163,244)(164,246)(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)(172,253)(173,255)(174,254)(175,259)(176,261)(177,260)(178,256)(179,258)(180,257)(181,262)(182,264)(183,263)(184,268)(185,270)(186,269)(187,265)(188,267)(189,266)(190,298)(191,300)(192,299)(193,304)(194,306)(195,305)(196,301)(197,303)(198,302)(199,307)(200,309)(201,308)(202,313)(203,315)(204,314)(205,310)(206,312)(207,311)(208,316)(209,318)(210,317)(211,322)(212,324)(213,323)(214,319)(215,321)(216,320)(217,271)(218,273)(219,272)(220,277)(221,279)(222,278)(223,274)(224,276)(225,275)(226,280)(227,282)(228,281)(229,286)(230,288)(231,287)(232,283)(233,285)(234,284)(235,289)(236,291)(237,290)(238,295)(239,297)(240,296)(241,292)(242,294)(243,293);;
s2 := ( 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)( 56, 57)( 58, 60)( 61, 62)( 64, 80)( 65, 79)( 66, 81)( 67, 73)( 68, 75)( 69, 74)( 70, 78)( 71, 77)( 72, 76)( 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)(137,138)(139,141)(142,143)(145,161)(146,160)(147,162)(148,154)(149,156)(150,155)(151,159)(152,158)(153,157)(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)(218,219)(220,222)(223,224)(226,242)(227,241)(228,243)(229,235)(230,237)(231,236)(232,240)(233,239)(234,238)(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)(299,300)(301,303)(304,305)(307,323)(308,322)(309,324)(310,316)(311,318)(312,317)(313,321)(314,320)(315,319);;
s3 := ( 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, 64)( 29, 66)( 30, 65)( 31, 67)( 32, 69)( 33, 68)( 34, 70)( 35, 72)( 36, 71)( 37, 55)( 38, 57)( 39, 56)( 40, 58)( 41, 60)( 42, 59)( 43, 61)( 44, 63)( 45, 62)( 46, 73)( 47, 75)( 48, 74)( 49, 76)( 50, 78)( 51, 77)( 52, 79)( 53, 81)( 54, 80)( 82, 91)( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)(107,108)(109,145)(110,147)(111,146)(112,148)(113,150)(114,149)(115,151)(116,153)(117,152)(118,136)(119,138)(120,137)(121,139)(122,141)(123,140)(124,142)(125,144)(126,143)(127,154)(128,156)(129,155)(130,157)(131,159)(132,158)(133,160)(134,162)(135,161)(163,172)(164,174)(165,173)(166,175)(167,177)(168,176)(169,178)(170,180)(171,179)(182,183)(185,186)(188,189)(190,226)(191,228)(192,227)(193,229)(194,231)(195,230)(196,232)(197,234)(198,233)(199,217)(200,219)(201,218)(202,220)(203,222)(204,221)(205,223)(206,225)(207,224)(208,235)(209,237)(210,236)(211,238)(212,240)(213,239)(214,241)(215,243)(216,242)(244,253)(245,255)(246,254)(247,256)(248,258)(249,257)(250,259)(251,261)(252,260)(263,264)(266,267)(269,270)(271,307)(272,309)(273,308)(274,310)(275,312)(276,311)(277,313)(278,315)(279,314)(280,298)(281,300)(282,299)(283,301)(284,303)(285,302)(286,304)(287,306)(288,305)(289,316)(290,318)(291,317)(292,319)(293,321)(294,320)(295,322)(296,324)(297,323);;
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,
s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2,
s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2 ];;
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)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 55)( 29, 57)( 30, 56)( 31, 61)( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)( 47, 75)( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(163,244)(164,246)(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)(172,253)(173,255)(174,254)(175,259)(176,261)(177,260)(178,256)(179,258)(180,257)(181,262)(182,264)(183,263)(184,268)(185,270)(186,269)(187,265)(188,267)(189,266)(190,298)(191,300)(192,299)(193,304)(194,306)(195,305)(196,301)(197,303)(198,302)(199,307)(200,309)(201,308)(202,313)(203,315)(204,314)(205,310)(206,312)(207,311)(208,316)(209,318)(210,317)(211,322)(212,324)(213,323)(214,319)(215,321)(216,320)(217,271)(218,273)(219,272)(220,277)(221,279)(222,278)(223,274)(224,276)(225,275)(226,280)(227,282)(228,281)(229,286)(230,288)(231,287)(232,283)(233,285)(234,284)(235,289)(236,291)(237,290)(238,295)(239,297)(240,296)(241,292)(242,294)(243,293);
s2 := 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)( 56, 57)( 58, 60)( 61, 62)( 64, 80)( 65, 79)( 66, 81)( 67, 73)( 68, 75)( 69, 74)( 70, 78)( 71, 77)( 72, 76)( 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)(137,138)(139,141)(142,143)(145,161)(146,160)(147,162)(148,154)(149,156)(150,155)(151,159)(152,158)(153,157)(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)(218,219)(220,222)(223,224)(226,242)(227,241)(228,243)(229,235)(230,237)(231,236)(232,240)(233,239)(234,238)(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)(299,300)(301,303)(304,305)(307,323)(308,322)(309,324)(310,316)(311,318)(312,317)(313,321)(314,320)(315,319);
s3 := 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, 64)( 29, 66)( 30, 65)( 31, 67)( 32, 69)( 33, 68)( 34, 70)( 35, 72)( 36, 71)( 37, 55)( 38, 57)( 39, 56)( 40, 58)( 41, 60)( 42, 59)( 43, 61)( 44, 63)( 45, 62)( 46, 73)( 47, 75)( 48, 74)( 49, 76)( 50, 78)( 51, 77)( 52, 79)( 53, 81)( 54, 80)( 82, 91)( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)(107,108)(109,145)(110,147)(111,146)(112,148)(113,150)(114,149)(115,151)(116,153)(117,152)(118,136)(119,138)(120,137)(121,139)(122,141)(123,140)(124,142)(125,144)(126,143)(127,154)(128,156)(129,155)(130,157)(131,159)(132,158)(133,160)(134,162)(135,161)(163,172)(164,174)(165,173)(166,175)(167,177)(168,176)(169,178)(170,180)(171,179)(182,183)(185,186)(188,189)(190,226)(191,228)(192,227)(193,229)(194,231)(195,230)(196,232)(197,234)(198,233)(199,217)(200,219)(201,218)(202,220)(203,222)(204,221)(205,223)(206,225)(207,224)(208,235)(209,237)(210,236)(211,238)(212,240)(213,239)(214,241)(215,243)(216,242)(244,253)(245,255)(246,254)(247,256)(248,258)(249,257)(250,259)(251,261)(252,260)(263,264)(266,267)(269,270)(271,307)(272,309)(273,308)(274,310)(275,312)(276,311)(277,313)(278,315)(279,314)(280,298)(281,300)(282,299)(283,301)(284,303)(285,302)(286,304)(287,306)(288,305)(289,316)(290,318)(291,317)(292,319)(293,321)(294,320)(295,322)(296,324)(297,323);
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,
s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2,
s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2 >;
References : None.
to this polytope