Polytope of Type {4,9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,9,6}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1789)
Rank : 4
Schlafli Type : {4,9,6}
Number of vertices, edges, etc : 4, 54, 81, 18
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Non-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) :
   3-fold quotients : {4,9,6}*432, {4,3,6}*432
   9-fold quotients : {4,9,2}*144, {4,3,6}*144
   27-fold quotients : {4,3,2}*48
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*s3*s2*s3*s2*s1*s3*s2*s3> of order 3.
      6 facets:
         6 of {4,9}*72
      4 vertex figures:
         4 of 3-fold non-regular quotient of {9,6}*324a
   P/N, where N=<s1*s2*s3*s2*s1*s2*s3*s2> of order 3.
      6 facets:
         6 of {4,9}*72
      4 vertex figures:
         4 of 3-fold non-regular quotient of {9,6}*324a

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