Polytope of Type {9,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,4}*1296a
if this polytope has a name.
Group : SmallGroup(1296,891)
Rank : 4
Schlafli Type : {9,6,4}
Number of vertices, edges, etc : 27, 81, 36, 4
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 : {9,6,2}*648a
   3-fold quotients : {9,6,4}*432, {3,6,4}*432a
   6-fold quotients : {9,6,2}*216, {3,6,2}*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.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      4 facets:
         4 of 3-fold non-regular quotient of {9,6}*324a
      15 vertex figures:
         6 of {6,4}*48a
         9 of {2,4}*16
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 3.
      4 facets:
         4 of 3-fold non-regular quotient of {9,6}*324a
      9 vertex figures:
         9 of {6,4}*48a

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