Polytope of Type {9,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,4,4}*1152a
if this polytope has a name.
Group : SmallGroup(1152,153992)
Rank : 4
Schlafli Type : {9,4,4}
Number of vertices, edges, etc : 9, 72, 32, 16
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 4
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) :
   2-fold quotients : {9,4,4}*576a
   3-fold quotients : {3,4,4}*384a
   6-fold quotients : {3,4,4}*192a
   8-fold quotients : {9,4,2}*144
   24-fold quotients : {3,4,2}*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*s1*s3*s2*s1> of order 2.
      8 facets:
         8 of {9,4}*72
      9 vertex figures:
         6 of 2-fold non-regular quotient of {4,4}*128
         3 of 2-fold non-regular quotient of {4,4}*128
   P/N, where N=<s1*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2> of order 2.
      8 facets:
         8 of {9,4}*72
      9 vertex figures:
         9 of 2-fold non-regular quotient of {4,4}*128
   P/N, where N=<s2*s3*s2*s3, s1*s2*s1*s2*s3*s2*s1*s3*s2*s1*s3> of order 4.
      4 facets:
         4 of {9,4}*72
      9 vertex figures:
         6 of 4-fold non-regular quotient of {4,4}*128
         3 of 4-fold non-regular quotient of {4,4}*128
   P/N, where N=<s1*s2*s3*s2*s1*s2*s3*s2, s2*s1*s2*s3*s2*s1*s3*s2> of order 4.
      4 facets:
         4 of {9,4}*72
      9 vertex figures:
         9 of 4-fold non-regular quotient of {4,4}*128

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