Polytope of Type {4,9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,9,6}*864
if this polytope has a name.
Group : SmallGroup(864,3999)
Rank : 4
Schlafli Type : {4,9,6}
Number of vertices, edges, etc : 8, 36, 54, 6
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,9,6,2} of size 1728
Vertex Figure Of :
   {2,4,9,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,9,6}*432
   3-fold quotients : {4,9,2}*288, {4,3,6}*288
   4-fold quotients : {2,9,6}*216
   6-fold quotients : {4,9,2}*144, {4,3,6}*144
   9-fold quotients : {4,3,2}*96
   12-fold quotients : {2,9,2}*72, {2,3,6}*72
   18-fold quotients : {4,3,2}*48
   36-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,9,6}*1728, {4,18,6}*1728b
Permutation Representation (GAP) :

s0 := (  1,111)(  2,112)(  3,109)(  4,110)(  5,115)(  6,116)(  7,113)(  8,114)
(  9,119)( 10,120)( 11,117)( 12,118)( 13,123)( 14,124)( 15,121)( 16,122)
( 17,127)( 18,128)( 19,125)( 20,126)( 21,131)( 22,132)( 23,129)( 24,130)
( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)( 32,138)
( 33,143)( 34,144)( 35,141)( 36,142)( 37,147)( 38,148)( 39,145)( 40,146)
( 41,151)( 42,152)( 43,149)( 44,150)( 45,155)( 46,156)( 47,153)( 48,154)
( 49,159)( 50,160)( 51,157)( 52,158)( 53,163)( 54,164)( 55,161)( 56,162)
( 57,167)( 58,168)( 59,165)( 60,166)( 61,171)( 62,172)( 63,169)( 64,170)
( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)( 72,178)
( 73,183)( 74,184)( 75,181)( 76,182)( 77,187)( 78,188)( 79,185)( 80,186)
( 81,191)( 82,192)( 83,189)( 84,190)( 85,195)( 86,196)( 87,193)( 88,194)
( 89,199)( 90,200)( 91,197)( 92,198)( 93,203)( 94,204)( 95,201)( 96,202)
( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)(104,210)
(105,215)(106,216)(107,213)(108,214);;
s1 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 25)( 14, 27)( 15, 26)
( 16, 28)( 17, 33)( 18, 35)( 19, 34)( 20, 36)( 21, 29)( 22, 31)( 23, 30)
( 24, 32)( 37, 77)( 38, 79)( 39, 78)( 40, 80)( 41, 73)( 42, 75)( 43, 74)
( 44, 76)( 45, 81)( 46, 83)( 47, 82)( 48, 84)( 49,101)( 50,103)( 51,102)
( 52,104)( 53, 97)( 54, 99)( 55, 98)( 56,100)( 57,105)( 58,107)( 59,106)
( 60,108)( 61, 89)( 62, 91)( 63, 90)( 64, 92)( 65, 85)( 66, 87)( 67, 86)
( 68, 88)( 69, 93)( 70, 95)( 71, 94)( 72, 96)(110,111)(113,117)(114,119)
(115,118)(116,120)(121,133)(122,135)(123,134)(124,136)(125,141)(126,143)
(127,142)(128,144)(129,137)(130,139)(131,138)(132,140)(145,185)(146,187)
(147,186)(148,188)(149,181)(150,183)(151,182)(152,184)(153,189)(154,191)
(155,190)(156,192)(157,209)(158,211)(159,210)(160,212)(161,205)(162,207)
(163,206)(164,208)(165,213)(166,215)(167,214)(168,216)(169,197)(170,199)
(171,198)(172,200)(173,193)(174,195)(175,194)(176,196)(177,201)(178,203)
(179,202)(180,204);;
s2 := (  1, 85)(  2, 88)(  3, 87)(  4, 86)(  5, 93)(  6, 96)(  7, 95)(  8, 94)
(  9, 89)( 10, 92)( 11, 91)( 12, 90)( 13, 73)( 14, 76)( 15, 75)( 16, 74)
( 17, 81)( 18, 84)( 19, 83)( 20, 82)( 21, 77)( 22, 80)( 23, 79)( 24, 78)
( 25, 97)( 26,100)( 27, 99)( 28, 98)( 29,105)( 30,108)( 31,107)( 32,106)
( 33,101)( 34,104)( 35,103)( 36,102)( 37, 49)( 38, 52)( 39, 51)( 40, 50)
( 41, 57)( 42, 60)( 43, 59)( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)
( 62, 64)( 65, 69)( 66, 72)( 67, 71)( 68, 70)(109,193)(110,196)(111,195)
(112,194)(113,201)(114,204)(115,203)(116,202)(117,197)(118,200)(119,199)
(120,198)(121,181)(122,184)(123,183)(124,182)(125,189)(126,192)(127,191)
(128,190)(129,185)(130,188)(131,187)(132,186)(133,205)(134,208)(135,207)
(136,206)(137,213)(138,216)(139,215)(140,214)(141,209)(142,212)(143,211)
(144,210)(145,157)(146,160)(147,159)(148,158)(149,165)(150,168)(151,167)
(152,166)(153,161)(154,164)(155,163)(156,162)(170,172)(173,177)(174,180)
(175,179)(176,178);;
s3 := ( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)
( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)
( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)
( 93,105)( 94,106)( 95,107)( 96,108)(121,133)(122,134)(123,135)(124,136)
(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)
(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)
(165,177)(166,178)(167,179)(168,180)(193,205)(194,206)(195,207)(196,208)
(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216);;
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, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
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(216)!(  1,111)(  2,112)(  3,109)(  4,110)(  5,115)(  6,116)(  7,113)
(  8,114)(  9,119)( 10,120)( 11,117)( 12,118)( 13,123)( 14,124)( 15,121)
( 16,122)( 17,127)( 18,128)( 19,125)( 20,126)( 21,131)( 22,132)( 23,129)
( 24,130)( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)
( 32,138)( 33,143)( 34,144)( 35,141)( 36,142)( 37,147)( 38,148)( 39,145)
( 40,146)( 41,151)( 42,152)( 43,149)( 44,150)( 45,155)( 46,156)( 47,153)
( 48,154)( 49,159)( 50,160)( 51,157)( 52,158)( 53,163)( 54,164)( 55,161)
( 56,162)( 57,167)( 58,168)( 59,165)( 60,166)( 61,171)( 62,172)( 63,169)
( 64,170)( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)
( 72,178)( 73,183)( 74,184)( 75,181)( 76,182)( 77,187)( 78,188)( 79,185)
( 80,186)( 81,191)( 82,192)( 83,189)( 84,190)( 85,195)( 86,196)( 87,193)
( 88,194)( 89,199)( 90,200)( 91,197)( 92,198)( 93,203)( 94,204)( 95,201)
( 96,202)( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)
(104,210)(105,215)(106,216)(107,213)(108,214);
s1 := Sym(216)!(  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 25)( 14, 27)
( 15, 26)( 16, 28)( 17, 33)( 18, 35)( 19, 34)( 20, 36)( 21, 29)( 22, 31)
( 23, 30)( 24, 32)( 37, 77)( 38, 79)( 39, 78)( 40, 80)( 41, 73)( 42, 75)
( 43, 74)( 44, 76)( 45, 81)( 46, 83)( 47, 82)( 48, 84)( 49,101)( 50,103)
( 51,102)( 52,104)( 53, 97)( 54, 99)( 55, 98)( 56,100)( 57,105)( 58,107)
( 59,106)( 60,108)( 61, 89)( 62, 91)( 63, 90)( 64, 92)( 65, 85)( 66, 87)
( 67, 86)( 68, 88)( 69, 93)( 70, 95)( 71, 94)( 72, 96)(110,111)(113,117)
(114,119)(115,118)(116,120)(121,133)(122,135)(123,134)(124,136)(125,141)
(126,143)(127,142)(128,144)(129,137)(130,139)(131,138)(132,140)(145,185)
(146,187)(147,186)(148,188)(149,181)(150,183)(151,182)(152,184)(153,189)
(154,191)(155,190)(156,192)(157,209)(158,211)(159,210)(160,212)(161,205)
(162,207)(163,206)(164,208)(165,213)(166,215)(167,214)(168,216)(169,197)
(170,199)(171,198)(172,200)(173,193)(174,195)(175,194)(176,196)(177,201)
(178,203)(179,202)(180,204);
s2 := Sym(216)!(  1, 85)(  2, 88)(  3, 87)(  4, 86)(  5, 93)(  6, 96)(  7, 95)
(  8, 94)(  9, 89)( 10, 92)( 11, 91)( 12, 90)( 13, 73)( 14, 76)( 15, 75)
( 16, 74)( 17, 81)( 18, 84)( 19, 83)( 20, 82)( 21, 77)( 22, 80)( 23, 79)
( 24, 78)( 25, 97)( 26,100)( 27, 99)( 28, 98)( 29,105)( 30,108)( 31,107)
( 32,106)( 33,101)( 34,104)( 35,103)( 36,102)( 37, 49)( 38, 52)( 39, 51)
( 40, 50)( 41, 57)( 42, 60)( 43, 59)( 44, 58)( 45, 53)( 46, 56)( 47, 55)
( 48, 54)( 62, 64)( 65, 69)( 66, 72)( 67, 71)( 68, 70)(109,193)(110,196)
(111,195)(112,194)(113,201)(114,204)(115,203)(116,202)(117,197)(118,200)
(119,199)(120,198)(121,181)(122,184)(123,183)(124,182)(125,189)(126,192)
(127,191)(128,190)(129,185)(130,188)(131,187)(132,186)(133,205)(134,208)
(135,207)(136,206)(137,213)(138,216)(139,215)(140,214)(141,209)(142,212)
(143,211)(144,210)(145,157)(146,160)(147,159)(148,158)(149,165)(150,168)
(151,167)(152,166)(153,161)(154,164)(155,163)(156,162)(170,172)(173,177)
(174,180)(175,179)(176,178);
s3 := Sym(216)!( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)
( 20, 32)( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 49, 61)( 50, 62)( 51, 63)
( 52, 64)( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)
( 60, 72)( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)
( 92,104)( 93,105)( 94,106)( 95,107)( 96,108)(121,133)(122,134)(123,135)
(124,136)(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)
(132,144)(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)
(164,176)(165,177)(166,178)(167,179)(168,180)(193,205)(194,206)(195,207)
(196,208)(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)
(204,216);
poly := sub<Sym(216)|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, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope