Polytope of Type {9,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,3}*972a
if this polytope has a name.
Group : SmallGroup(972,108)
Rank : 4
Schlafli Type : {9,6,3}
Number of vertices, edges, etc : 27, 81, 27, 3
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}*324b
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}*1944a, {18,6,3}*1944a
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s1*s2*s1*s2> of order 3.
3 facets:
3 of 3-fold non-regular quotient of {9,6}*324a
15 vertex figures:
9 of {2,3}*12
6 of {6,3}*36
P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
3 facets:
3 of 3-fold non-regular quotient of {9,6}*324a
9 vertex figures:
9 of {6,3}*36
Permutation Representation (GAP) :
s0 := ( 4, 8)( 5, 9)( 6, 7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 82,181)( 83,182)( 84,183)( 85,188)( 86,189)( 87,187)( 88,186)( 89,184)( 90,185)( 91,172)( 92,173)( 93,174)( 94,179)( 95,180)( 96,178)( 97,177)( 98,175)( 99,176)(100,163)(101,164)(102,165)(103,170)(104,171)(105,169)(106,168)(107,166)(108,167)(109,208)(110,209)(111,210)(112,215)(113,216)(114,214)(115,213)(116,211)(117,212)(118,199)(119,200)(120,201)(121,206)(122,207)(123,205)(124,204)(125,202)(126,203)(127,190)(128,191)(129,192)(130,197)(131,198)(132,196)(133,195)(134,193)(135,194)(136,235)(137,236)(138,237)(139,242)(140,243)(141,241)(142,240)(143,238)(144,239)(145,226)(146,227)(147,228)(148,233)(149,234)(150,232)(151,231)(152,229)(153,230)(154,217)(155,218)(156,219)(157,224)(158,225)(159,223)(160,222)(161,220)(162,221);;
s1 := ( 1, 82)( 2, 83)( 3, 84)( 4, 88)( 5, 89)( 6, 90)( 7, 85)( 8, 86)( 9, 87)( 10,100)( 11,101)( 12,102)( 13,106)( 14,107)( 15,108)( 16,103)( 17,104)( 18,105)( 19, 91)( 20, 92)( 21, 93)( 22, 97)( 23, 98)( 24, 99)( 25, 94)( 26, 95)( 27, 96)( 28,112)( 29,113)( 30,114)( 31,109)( 32,110)( 33,111)( 34,115)( 35,116)( 36,117)( 37,130)( 38,131)( 39,132)( 40,127)( 41,128)( 42,129)( 43,133)( 44,134)( 45,135)( 46,121)( 47,122)( 48,123)( 49,118)( 50,119)( 51,120)( 52,124)( 53,125)( 54,126)( 55,142)( 56,143)( 57,144)( 58,139)( 59,140)( 60,141)( 61,136)( 62,137)( 63,138)( 64,160)( 65,161)( 66,162)( 67,157)( 68,158)( 69,159)( 70,154)( 71,155)( 72,156)( 73,151)( 74,152)( 75,153)( 76,148)( 77,149)( 78,150)( 79,145)( 80,146)( 81,147)(163,181)(164,182)(165,183)(166,187)(167,188)(168,189)(169,184)(170,185)(171,186)(175,178)(176,179)(177,180)(190,211)(191,212)(192,213)(193,208)(194,209)(195,210)(196,214)(197,215)(198,216)(199,202)(200,203)(201,204)(217,241)(218,242)(219,243)(220,238)(221,239)(222,240)(223,235)(224,236)(225,237)(226,232)(227,233)(228,234);;
s2 := ( 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)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 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)(137,138)(139,142)(140,144)(141,143)(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)(159,161)(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)(218,219)(220,223)(221,225)(222,224)(227,228)(229,232)(230,234)(231,233)(236,237)(238,241)(239,243)(240,242);;
s3 := ( 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)(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)(195,224)(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)(203,234)(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)(211,241)(212,243)(213,242)(214,238)(215,240)(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,
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 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)!( 4, 8)( 5, 9)( 6, 7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 82,181)( 83,182)( 84,183)( 85,188)( 86,189)( 87,187)( 88,186)( 89,184)( 90,185)( 91,172)( 92,173)( 93,174)( 94,179)( 95,180)( 96,178)( 97,177)( 98,175)( 99,176)(100,163)(101,164)(102,165)(103,170)(104,171)(105,169)(106,168)(107,166)(108,167)(109,208)(110,209)(111,210)(112,215)(113,216)(114,214)(115,213)(116,211)(117,212)(118,199)(119,200)(120,201)(121,206)(122,207)(123,205)(124,204)(125,202)(126,203)(127,190)(128,191)(129,192)(130,197)(131,198)(132,196)(133,195)(134,193)(135,194)(136,235)(137,236)(138,237)(139,242)(140,243)(141,241)(142,240)(143,238)(144,239)(145,226)(146,227)(147,228)(148,233)(149,234)(150,232)(151,231)(152,229)(153,230)(154,217)(155,218)(156,219)(157,224)(158,225)(159,223)(160,222)(161,220)(162,221);
s1 := Sym(243)!( 1, 82)( 2, 83)( 3, 84)( 4, 88)( 5, 89)( 6, 90)( 7, 85)( 8, 86)( 9, 87)( 10,100)( 11,101)( 12,102)( 13,106)( 14,107)( 15,108)( 16,103)( 17,104)( 18,105)( 19, 91)( 20, 92)( 21, 93)( 22, 97)( 23, 98)( 24, 99)( 25, 94)( 26, 95)( 27, 96)( 28,112)( 29,113)( 30,114)( 31,109)( 32,110)( 33,111)( 34,115)( 35,116)( 36,117)( 37,130)( 38,131)( 39,132)( 40,127)( 41,128)( 42,129)( 43,133)( 44,134)( 45,135)( 46,121)( 47,122)( 48,123)( 49,118)( 50,119)( 51,120)( 52,124)( 53,125)( 54,126)( 55,142)( 56,143)( 57,144)( 58,139)( 59,140)( 60,141)( 61,136)( 62,137)( 63,138)( 64,160)( 65,161)( 66,162)( 67,157)( 68,158)( 69,159)( 70,154)( 71,155)( 72,156)( 73,151)( 74,152)( 75,153)( 76,148)( 77,149)( 78,150)( 79,145)( 80,146)( 81,147)(163,181)(164,182)(165,183)(166,187)(167,188)(168,189)(169,184)(170,185)(171,186)(175,178)(176,179)(177,180)(190,211)(191,212)(192,213)(193,208)(194,209)(195,210)(196,214)(197,215)(198,216)(199,202)(200,203)(201,204)(217,241)(218,242)(219,243)(220,238)(221,239)(222,240)(223,235)(224,236)(225,237)(226,232)(227,233)(228,234);
s2 := 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)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 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)(137,138)(139,142)(140,144)(141,143)(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)(159,161)(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)(218,219)(220,223)(221,225)(222,224)(227,228)(229,232)(230,234)(231,233)(236,237)(238,241)(239,243)(240,242);
s3 := 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, 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)(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)(195,224)(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)(203,234)(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)(211,241)(212,243)(213,242)(214,238)(215,240)(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, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2,
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