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Polytope of Type {9,6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,9}*972
if this polytope has a name.
Group : SmallGroup(972,99)
Rank : 4
Schlafli Type : {9,6,9}
Number of vertices, edges, etc : 9, 27, 27, 9
Order of s0s1s2s3 : 9
Order of s0s1s2s3s2s1 : 6
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {9,6,9,2} of size 1944
Vertex Figure Of :
   {2,9,6,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {9,2,9}*324, {3,6,9}*324, {9,6,3}*324
   9-fold quotients : {3,2,9}*108, {9,2,3}*108, {3,6,3}*108
   27-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
   2-fold covers : {9,6,18}*1944a, {18,6,9}*1944a
Permutation Representation (GAP) :
s0 := (  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, 61)( 29, 63)( 30, 62)( 31, 58)
( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 70)( 38, 72)( 39, 71)
( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 46, 79)( 47, 81)
( 48, 80)( 49, 76)( 50, 78)( 51, 77)( 52, 73)( 53, 75)( 54, 74)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,142)(110,144)(111,143)(112,139)(113,141)
(114,140)(115,136)(116,138)(117,137)(118,151)(119,153)(120,152)(121,148)
(122,150)(123,149)(124,145)(125,147)(126,146)(127,160)(128,162)(129,161)
(130,157)(131,159)(132,158)(133,154)(134,156)(135,155)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,223)(191,225)(192,224)(193,220)(194,222)(195,221)
(196,217)(197,219)(198,218)(199,232)(200,234)(201,233)(202,229)(203,231)
(204,230)(205,226)(206,228)(207,227)(208,241)(209,243)(210,242)(211,238)
(212,240)(213,239)(214,235)(215,237)(216,236);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)(  8, 33)
(  9, 32)( 10, 37)( 11, 39)( 12, 38)( 13, 43)( 14, 45)( 15, 44)( 16, 40)
( 17, 42)( 18, 41)( 19, 46)( 20, 48)( 21, 47)( 22, 52)( 23, 54)( 24, 53)
( 25, 49)( 26, 51)( 27, 50)( 55, 61)( 56, 63)( 57, 62)( 59, 60)( 64, 70)
( 65, 72)( 66, 71)( 68, 69)( 73, 79)( 74, 81)( 75, 80)( 77, 78)( 82,109)
( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)( 90,113)
( 91,118)( 92,120)( 93,119)( 94,124)( 95,126)( 96,125)( 97,121)( 98,123)
( 99,122)(100,127)(101,129)(102,128)(103,133)(104,135)(105,134)(106,130)
(107,132)(108,131)(136,142)(137,144)(138,143)(140,141)(145,151)(146,153)
(147,152)(149,150)(154,160)(155,162)(156,161)(158,159)(163,190)(164,192)
(165,191)(166,196)(167,198)(168,197)(169,193)(170,195)(171,194)(172,199)
(173,201)(174,200)(175,205)(176,207)(177,206)(178,202)(179,204)(180,203)
(181,208)(182,210)(183,209)(184,214)(185,216)(186,215)(187,211)(188,213)
(189,212)(217,223)(218,225)(219,224)(221,222)(226,232)(227,234)(228,233)
(230,231)(235,241)(236,243)(237,242)(239,240);;
s2 := (  1, 82)(  2, 84)(  3, 83)(  4, 85)(  5, 87)(  6, 86)(  7, 88)(  8, 90)
(  9, 89)( 10,100)( 11,102)( 12,101)( 13,103)( 14,105)( 15,104)( 16,106)
( 17,108)( 18,107)( 19, 91)( 20, 93)( 21, 92)( 22, 94)( 23, 96)( 24, 95)
( 25, 97)( 26, 99)( 27, 98)( 28,110)( 29,109)( 30,111)( 31,113)( 32,112)
( 33,114)( 34,116)( 35,115)( 36,117)( 37,128)( 38,127)( 39,129)( 40,131)
( 41,130)( 42,132)( 43,134)( 44,133)( 45,135)( 46,119)( 47,118)( 48,120)
( 49,122)( 50,121)( 51,123)( 52,125)( 53,124)( 54,126)( 55,138)( 56,137)
( 57,136)( 58,141)( 59,140)( 60,139)( 61,144)( 62,143)( 63,142)( 64,156)
( 65,155)( 66,154)( 67,159)( 68,158)( 69,157)( 70,162)( 71,161)( 72,160)
( 73,147)( 74,146)( 75,145)( 76,150)( 77,149)( 78,148)( 79,153)( 80,152)
( 81,151)(163,181)(164,183)(165,182)(166,184)(167,186)(168,185)(169,187)
(170,189)(171,188)(173,174)(176,177)(179,180)(190,209)(191,208)(192,210)
(193,212)(194,211)(195,213)(196,215)(197,214)(198,216)(199,200)(202,203)
(205,206)(217,237)(218,236)(219,235)(220,240)(221,239)(222,238)(223,243)
(224,242)(225,241)(226,228)(229,231)(232,234);;
s3 := (  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)( 82,181)( 83,183)( 84,182)( 85,184)
( 86,186)( 87,185)( 88,187)( 89,189)( 90,188)( 91,172)( 92,174)( 93,173)
( 94,175)( 95,177)( 96,176)( 97,178)( 98,180)( 99,179)(100,163)(101,165)
(102,164)(103,166)(104,168)(105,167)(106,169)(107,171)(108,170)(109,208)
(110,210)(111,209)(112,211)(113,213)(114,212)(115,214)(116,216)(117,215)
(118,199)(119,201)(120,200)(121,202)(122,204)(123,203)(124,205)(125,207)
(126,206)(127,190)(128,192)(129,191)(130,193)(131,195)(132,194)(133,196)
(134,198)(135,197)(136,235)(137,237)(138,236)(139,238)(140,240)(141,239)
(142,241)(143,243)(144,242)(145,226)(146,228)(147,227)(148,229)(149,231)
(150,230)(151,232)(152,234)(153,233)(154,217)(155,219)(156,218)(157,220)
(158,222)(159,221)(160,223)(161,225)(162,224);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(243)!(  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, 61)( 29, 63)( 30, 62)
( 31, 58)( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 70)( 38, 72)
( 39, 71)( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 46, 79)
( 47, 81)( 48, 80)( 49, 76)( 50, 78)( 51, 77)( 52, 73)( 53, 75)( 54, 74)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,142)(110,144)(111,143)(112,139)
(113,141)(114,140)(115,136)(116,138)(117,137)(118,151)(119,153)(120,152)
(121,148)(122,150)(123,149)(124,145)(125,147)(126,146)(127,160)(128,162)
(129,161)(130,157)(131,159)(132,158)(133,154)(134,156)(135,155)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(190,223)(191,225)(192,224)(193,220)(194,222)
(195,221)(196,217)(197,219)(198,218)(199,232)(200,234)(201,233)(202,229)
(203,231)(204,230)(205,226)(206,228)(207,227)(208,241)(209,243)(210,242)
(211,238)(212,240)(213,239)(214,235)(215,237)(216,236);
s1 := Sym(243)!(  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)
(  8, 33)(  9, 32)( 10, 37)( 11, 39)( 12, 38)( 13, 43)( 14, 45)( 15, 44)
( 16, 40)( 17, 42)( 18, 41)( 19, 46)( 20, 48)( 21, 47)( 22, 52)( 23, 54)
( 24, 53)( 25, 49)( 26, 51)( 27, 50)( 55, 61)( 56, 63)( 57, 62)( 59, 60)
( 64, 70)( 65, 72)( 66, 71)( 68, 69)( 73, 79)( 74, 81)( 75, 80)( 77, 78)
( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)
( 90,113)( 91,118)( 92,120)( 93,119)( 94,124)( 95,126)( 96,125)( 97,121)
( 98,123)( 99,122)(100,127)(101,129)(102,128)(103,133)(104,135)(105,134)
(106,130)(107,132)(108,131)(136,142)(137,144)(138,143)(140,141)(145,151)
(146,153)(147,152)(149,150)(154,160)(155,162)(156,161)(158,159)(163,190)
(164,192)(165,191)(166,196)(167,198)(168,197)(169,193)(170,195)(171,194)
(172,199)(173,201)(174,200)(175,205)(176,207)(177,206)(178,202)(179,204)
(180,203)(181,208)(182,210)(183,209)(184,214)(185,216)(186,215)(187,211)
(188,213)(189,212)(217,223)(218,225)(219,224)(221,222)(226,232)(227,234)
(228,233)(230,231)(235,241)(236,243)(237,242)(239,240);
s2 := Sym(243)!(  1, 82)(  2, 84)(  3, 83)(  4, 85)(  5, 87)(  6, 86)(  7, 88)
(  8, 90)(  9, 89)( 10,100)( 11,102)( 12,101)( 13,103)( 14,105)( 15,104)
( 16,106)( 17,108)( 18,107)( 19, 91)( 20, 93)( 21, 92)( 22, 94)( 23, 96)
( 24, 95)( 25, 97)( 26, 99)( 27, 98)( 28,110)( 29,109)( 30,111)( 31,113)
( 32,112)( 33,114)( 34,116)( 35,115)( 36,117)( 37,128)( 38,127)( 39,129)
( 40,131)( 41,130)( 42,132)( 43,134)( 44,133)( 45,135)( 46,119)( 47,118)
( 48,120)( 49,122)( 50,121)( 51,123)( 52,125)( 53,124)( 54,126)( 55,138)
( 56,137)( 57,136)( 58,141)( 59,140)( 60,139)( 61,144)( 62,143)( 63,142)
( 64,156)( 65,155)( 66,154)( 67,159)( 68,158)( 69,157)( 70,162)( 71,161)
( 72,160)( 73,147)( 74,146)( 75,145)( 76,150)( 77,149)( 78,148)( 79,153)
( 80,152)( 81,151)(163,181)(164,183)(165,182)(166,184)(167,186)(168,185)
(169,187)(170,189)(171,188)(173,174)(176,177)(179,180)(190,209)(191,208)
(192,210)(193,212)(194,211)(195,213)(196,215)(197,214)(198,216)(199,200)
(202,203)(205,206)(217,237)(218,236)(219,235)(220,240)(221,239)(222,238)
(223,243)(224,242)(225,241)(226,228)(229,231)(232,234);
s3 := Sym(243)!(  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)( 82,181)( 83,183)( 84,182)
( 85,184)( 86,186)( 87,185)( 88,187)( 89,189)( 90,188)( 91,172)( 92,174)
( 93,173)( 94,175)( 95,177)( 96,176)( 97,178)( 98,180)( 99,179)(100,163)
(101,165)(102,164)(103,166)(104,168)(105,167)(106,169)(107,171)(108,170)
(109,208)(110,210)(111,209)(112,211)(113,213)(114,212)(115,214)(116,216)
(117,215)(118,199)(119,201)(120,200)(121,202)(122,204)(123,203)(124,205)
(125,207)(126,206)(127,190)(128,192)(129,191)(130,193)(131,195)(132,194)
(133,196)(134,198)(135,197)(136,235)(137,237)(138,236)(139,238)(140,240)
(141,239)(142,241)(143,243)(144,242)(145,226)(146,228)(147,227)(148,229)
(149,231)(150,230)(151,232)(152,234)(153,233)(154,217)(155,219)(156,218)
(157,220)(158,222)(159,221)(160,223)(161,225)(162,224);
poly := sub<Sym(243)|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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
to this polytope