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Polytope of Type {9,6,8,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,8,2}*1728
if this polytope has a name.
Group : SmallGroup(1728,17171)
Rank : 5
Schlafli Type : {9,6,8,2}
Number of vertices, edges, etc : 9, 27, 24, 8, 2
Order of s0s1s2s3s4 : 72
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   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 : {9,6,4,2}*864
   3-fold quotients : {9,2,8,2}*576, {3,6,8,2}*576
   4-fold quotients : {9,6,2,2}*432
   6-fold quotients : {9,2,4,2}*288, {3,6,4,2}*288
   9-fold quotients : {3,2,8,2}*192
   12-fold quotients : {9,2,2,2}*144, {3,6,2,2}*144
   18-fold quotients : {3,2,4,2}*96
   36-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 21)( 11, 20)( 12, 19)( 13, 27)
( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 48)( 38, 47)( 39, 46)( 40, 54)( 41, 53)( 42, 52)( 43, 51)
( 44, 50)( 45, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 75)( 65, 74)
( 66, 73)( 67, 81)( 68, 80)( 69, 79)( 70, 78)( 71, 77)( 72, 76)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 91,102)( 92,101)( 93,100)( 94,108)( 95,107)
( 96,106)( 97,105)( 98,104)( 99,103)(110,111)(112,115)(113,117)(114,116)
(118,129)(119,128)(120,127)(121,135)(122,134)(123,133)(124,132)(125,131)
(126,130)(137,138)(139,142)(140,144)(141,143)(145,156)(146,155)(147,154)
(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(164,165)(166,169)
(167,171)(168,170)(172,183)(173,182)(174,181)(175,189)(176,188)(177,187)
(178,186)(179,185)(180,184)(191,192)(193,196)(194,198)(195,197)(199,210)
(200,209)(201,208)(202,216)(203,215)(204,214)(205,213)(206,212)(207,211);;
s1 := (  1, 13)(  2, 15)(  3, 14)(  4, 10)(  5, 12)(  6, 11)(  7, 16)(  8, 18)
(  9, 17)( 19, 24)( 20, 23)( 21, 22)( 25, 27)( 28, 40)( 29, 42)( 30, 41)
( 31, 37)( 32, 39)( 33, 38)( 34, 43)( 35, 45)( 36, 44)( 46, 51)( 47, 50)
( 48, 49)( 52, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)
( 61, 70)( 62, 72)( 63, 71)( 73, 78)( 74, 77)( 75, 76)( 79, 81)( 82, 94)
( 83, 96)( 84, 95)( 85, 91)( 86, 93)( 87, 92)( 88, 97)( 89, 99)( 90, 98)
(100,105)(101,104)(102,103)(106,108)(109,121)(110,123)(111,122)(112,118)
(113,120)(114,119)(115,124)(116,126)(117,125)(127,132)(128,131)(129,130)
(133,135)(136,148)(137,150)(138,149)(139,145)(140,147)(141,146)(142,151)
(143,153)(144,152)(154,159)(155,158)(156,157)(160,162)(163,175)(164,177)
(165,176)(166,172)(167,174)(168,173)(169,178)(170,180)(171,179)(181,186)
(182,185)(183,184)(187,189)(190,202)(191,204)(192,203)(193,199)(194,201)
(195,200)(196,205)(197,207)(198,206)(208,213)(209,212)(210,211)(214,216);;
s2 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)
( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)( 68, 98)
( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)( 76,106)
( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(109,163)(110,164)(111,165)
(112,169)(113,170)(114,171)(115,166)(116,167)(117,168)(118,172)(119,173)
(120,174)(121,178)(122,179)(123,180)(124,175)(125,176)(126,177)(127,181)
(128,182)(129,183)(130,187)(131,188)(132,189)(133,184)(134,185)(135,186)
(136,190)(137,191)(138,192)(139,196)(140,197)(141,198)(142,193)(143,194)
(144,195)(145,199)(146,200)(147,201)(148,205)(149,206)(150,207)(151,202)
(152,203)(153,204)(154,208)(155,209)(156,210)(157,214)(158,215)(159,216)
(160,211)(161,212)(162,213);;
s3 := (  1,109)(  2,110)(  3,111)(  4,112)(  5,113)(  6,114)(  7,115)(  8,116)
(  9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)
( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)
( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)
( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)
( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)( 56,191)
( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)
( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)
( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)
( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)
( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)
( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)
(105,186)(106,187)(107,188)(108,189);;
s4 := (217,218);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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(218)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 21)( 11, 20)( 12, 19)
( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 34)
( 32, 36)( 33, 35)( 37, 48)( 38, 47)( 39, 46)( 40, 54)( 41, 53)( 42, 52)
( 43, 51)( 44, 50)( 45, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 75)
( 65, 74)( 66, 73)( 67, 81)( 68, 80)( 69, 79)( 70, 78)( 71, 77)( 72, 76)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 91,102)( 92,101)( 93,100)( 94,108)
( 95,107)( 96,106)( 97,105)( 98,104)( 99,103)(110,111)(112,115)(113,117)
(114,116)(118,129)(119,128)(120,127)(121,135)(122,134)(123,133)(124,132)
(125,131)(126,130)(137,138)(139,142)(140,144)(141,143)(145,156)(146,155)
(147,154)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(164,165)
(166,169)(167,171)(168,170)(172,183)(173,182)(174,181)(175,189)(176,188)
(177,187)(178,186)(179,185)(180,184)(191,192)(193,196)(194,198)(195,197)
(199,210)(200,209)(201,208)(202,216)(203,215)(204,214)(205,213)(206,212)
(207,211);
s1 := Sym(218)!(  1, 13)(  2, 15)(  3, 14)(  4, 10)(  5, 12)(  6, 11)(  7, 16)
(  8, 18)(  9, 17)( 19, 24)( 20, 23)( 21, 22)( 25, 27)( 28, 40)( 29, 42)
( 30, 41)( 31, 37)( 32, 39)( 33, 38)( 34, 43)( 35, 45)( 36, 44)( 46, 51)
( 47, 50)( 48, 49)( 52, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)
( 60, 65)( 61, 70)( 62, 72)( 63, 71)( 73, 78)( 74, 77)( 75, 76)( 79, 81)
( 82, 94)( 83, 96)( 84, 95)( 85, 91)( 86, 93)( 87, 92)( 88, 97)( 89, 99)
( 90, 98)(100,105)(101,104)(102,103)(106,108)(109,121)(110,123)(111,122)
(112,118)(113,120)(114,119)(115,124)(116,126)(117,125)(127,132)(128,131)
(129,130)(133,135)(136,148)(137,150)(138,149)(139,145)(140,147)(141,146)
(142,151)(143,153)(144,152)(154,159)(155,158)(156,157)(160,162)(163,175)
(164,177)(165,176)(166,172)(167,174)(168,173)(169,178)(170,180)(171,179)
(181,186)(182,185)(183,184)(187,189)(190,202)(191,204)(192,203)(193,199)
(194,201)(195,200)(196,205)(197,207)(198,206)(208,213)(209,212)(210,211)
(214,216);
s2 := Sym(218)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)
( 60, 90)( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)
( 68, 98)( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)
( 76,106)( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(109,163)(110,164)
(111,165)(112,169)(113,170)(114,171)(115,166)(116,167)(117,168)(118,172)
(119,173)(120,174)(121,178)(122,179)(123,180)(124,175)(125,176)(126,177)
(127,181)(128,182)(129,183)(130,187)(131,188)(132,189)(133,184)(134,185)
(135,186)(136,190)(137,191)(138,192)(139,196)(140,197)(141,198)(142,193)
(143,194)(144,195)(145,199)(146,200)(147,201)(148,205)(149,206)(150,207)
(151,202)(152,203)(153,204)(154,208)(155,209)(156,210)(157,214)(158,215)
(159,216)(160,211)(161,212)(162,213);
s3 := Sym(218)!(  1,109)(  2,110)(  3,111)(  4,112)(  5,113)(  6,114)(  7,115)
(  8,116)(  9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)
( 16,124)( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)
( 24,132)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)
( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)
( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)
( 56,191)( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)
( 64,199)( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)
( 72,207)( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)
( 80,215)( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)
( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)
( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)
(104,185)(105,186)(106,187)(107,188)(108,189);
s4 := Sym(218)!(217,218);
poly := sub<Sym(218)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s3*s4*s3*s4, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 

to this polytope