Polytope of Type {9,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,4}*1296c
if this polytope has a name.
Group : SmallGroup(1296,917)
Rank : 4
Schlafli Type : {9,6,4}
Number of vertices, edges, etc : 27, 81, 36, 4
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 : {9,6,2}*648b
   3-fold quotients : {3,6,4}*432a
   6-fold quotients : {3,6,2}*216
   9-fold quotients : {3,6,4}*144
   18-fold quotients : {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*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}*324b
      9 vertex figures:
         9 of {6,4}*48a
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
      4 facets:
         4 of 3-fold non-regular quotient of {9,6}*324b
      15 vertex figures:
         6 of {6,4}*48a
         9 of {2,4}*16

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