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