Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,4,9,6}

Atlas Canonical Name {2,4,9,6}*1728

Overview

Group
SmallGroup(1728,46115)
Rank
5
Schläfli Type
{2,4,9,6}
Vertices, edges, …
2, 8, 36, 54, 6
Order of s0s1s2s3s4
18
Order of s0s1s2s3s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

9-fold

12-fold

18-fold

36-fold

Covers minimal covers in bold

None in this atlas.

Representations

Permutation Representation (GAP) 
s0 := (1,2);;
s1 := (  3,113)(  4,114)(  5,111)(  6,112)(  7,117)(  8,118)(  9,115)( 10,116)( 11,121)( 12,122)( 13,119)( 14,120)( 15,125)( 16,126)( 17,123)( 18,124)( 19,129)( 20,130)( 21,127)( 22,128)( 23,133)( 24,134)( 25,131)( 26,132)( 27,137)( 28,138)( 29,135)( 30,136)( 31,141)( 32,142)( 33,139)( 34,140)( 35,145)( 36,146)( 37,143)( 38,144)( 39,149)( 40,150)( 41,147)( 42,148)( 43,153)( 44,154)( 45,151)( 46,152)( 47,157)( 48,158)( 49,155)( 50,156)( 51,161)( 52,162)( 53,159)( 54,160)( 55,165)( 56,166)( 57,163)( 58,164)( 59,169)( 60,170)( 61,167)( 62,168)( 63,173)( 64,174)( 65,171)( 66,172)( 67,177)( 68,178)( 69,175)( 70,176)( 71,181)( 72,182)( 73,179)( 74,180)( 75,185)( 76,186)( 77,183)( 78,184)( 79,189)( 80,190)( 81,187)( 82,188)( 83,193)( 84,194)( 85,191)( 86,192)( 87,197)( 88,198)( 89,195)( 90,196)( 91,201)( 92,202)( 93,199)( 94,200)( 95,205)( 96,206)( 97,203)( 98,204)( 99,209)(100,210)(101,207)(102,208)(103,213)(104,214)(105,211)(106,212)(107,217)(108,218)(109,215)(110,216);;
s2 := (  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 27)( 16, 29)( 17, 28)( 18, 30)( 19, 35)( 20, 37)( 21, 36)( 22, 38)( 23, 31)( 24, 33)( 25, 32)( 26, 34)( 39, 79)( 40, 81)( 41, 80)( 42, 82)( 43, 75)( 44, 77)( 45, 76)( 46, 78)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51,103)( 52,105)( 53,104)( 54,106)( 55, 99)( 56,101)( 57,100)( 58,102)( 59,107)( 60,109)( 61,108)( 62,110)( 63, 91)( 64, 93)( 65, 92)( 66, 94)( 67, 87)( 68, 89)( 69, 88)( 70, 90)( 71, 95)( 72, 97)( 73, 96)( 74, 98)(112,113)(115,119)(116,121)(117,120)(118,122)(123,135)(124,137)(125,136)(126,138)(127,143)(128,145)(129,144)(130,146)(131,139)(132,141)(133,140)(134,142)(147,187)(148,189)(149,188)(150,190)(151,183)(152,185)(153,184)(154,186)(155,191)(156,193)(157,192)(158,194)(159,211)(160,213)(161,212)(162,214)(163,207)(164,209)(165,208)(166,210)(167,215)(168,217)(169,216)(170,218)(171,199)(172,201)(173,200)(174,202)(175,195)(176,197)(177,196)(178,198)(179,203)(180,205)(181,204)(182,206);;
s3 := (  3, 87)(  4, 90)(  5, 89)(  6, 88)(  7, 95)(  8, 98)(  9, 97)( 10, 96)( 11, 91)( 12, 94)( 13, 93)( 14, 92)( 15, 75)( 16, 78)( 17, 77)( 18, 76)( 19, 83)( 20, 86)( 21, 85)( 22, 84)( 23, 79)( 24, 82)( 25, 81)( 26, 80)( 27, 99)( 28,102)( 29,101)( 30,100)( 31,107)( 32,110)( 33,109)( 34,108)( 35,103)( 36,106)( 37,105)( 38,104)( 39, 51)( 40, 54)( 41, 53)( 42, 52)( 43, 59)( 44, 62)( 45, 61)( 46, 60)( 47, 55)( 48, 58)( 49, 57)( 50, 56)( 64, 66)( 67, 71)( 68, 74)( 69, 73)( 70, 72)(111,195)(112,198)(113,197)(114,196)(115,203)(116,206)(117,205)(118,204)(119,199)(120,202)(121,201)(122,200)(123,183)(124,186)(125,185)(126,184)(127,191)(128,194)(129,193)(130,192)(131,187)(132,190)(133,189)(134,188)(135,207)(136,210)(137,209)(138,208)(139,215)(140,218)(141,217)(142,216)(143,211)(144,214)(145,213)(146,212)(147,159)(148,162)(149,161)(150,160)(151,167)(152,170)(153,169)(154,168)(155,163)(156,166)(157,165)(158,164)(172,174)(175,179)(176,182)(177,181)(178,180);;
s4 := ( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 25, 37)( 26, 38)( 51, 63)( 52, 64)( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 61, 73)( 62, 74)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)( 94,106)( 95,107)( 96,108)( 97,109)( 98,110)(123,135)(124,136)(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)(133,145)(134,146)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)(165,177)(166,178)(167,179)(168,180)(169,181)(170,182)(195,207)(196,208)(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)(206,218);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s1*s2*s1*s2*s1*s2*s1*s2, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s4*s3*s2*s3*s2*s3*s4*s3, 
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(218)!(1,2);
s1 := Sym(218)!(  3,113)(  4,114)(  5,111)(  6,112)(  7,117)(  8,118)(  9,115)( 10,116)( 11,121)( 12,122)( 13,119)( 14,120)( 15,125)( 16,126)( 17,123)( 18,124)( 19,129)( 20,130)( 21,127)( 22,128)( 23,133)( 24,134)( 25,131)( 26,132)( 27,137)( 28,138)( 29,135)( 30,136)( 31,141)( 32,142)( 33,139)( 34,140)( 35,145)( 36,146)( 37,143)( 38,144)( 39,149)( 40,150)( 41,147)( 42,148)( 43,153)( 44,154)( 45,151)( 46,152)( 47,157)( 48,158)( 49,155)( 50,156)( 51,161)( 52,162)( 53,159)( 54,160)( 55,165)( 56,166)( 57,163)( 58,164)( 59,169)( 60,170)( 61,167)( 62,168)( 63,173)( 64,174)( 65,171)( 66,172)( 67,177)( 68,178)( 69,175)( 70,176)( 71,181)( 72,182)( 73,179)( 74,180)( 75,185)( 76,186)( 77,183)( 78,184)( 79,189)( 80,190)( 81,187)( 82,188)( 83,193)( 84,194)( 85,191)( 86,192)( 87,197)( 88,198)( 89,195)( 90,196)( 91,201)( 92,202)( 93,199)( 94,200)( 95,205)( 96,206)( 97,203)( 98,204)( 99,209)(100,210)(101,207)(102,208)(103,213)(104,214)(105,211)(106,212)(107,217)(108,218)(109,215)(110,216);
s2 := Sym(218)!(  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 27)( 16, 29)( 17, 28)( 18, 30)( 19, 35)( 20, 37)( 21, 36)( 22, 38)( 23, 31)( 24, 33)( 25, 32)( 26, 34)( 39, 79)( 40, 81)( 41, 80)( 42, 82)( 43, 75)( 44, 77)( 45, 76)( 46, 78)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51,103)( 52,105)( 53,104)( 54,106)( 55, 99)( 56,101)( 57,100)( 58,102)( 59,107)( 60,109)( 61,108)( 62,110)( 63, 91)( 64, 93)( 65, 92)( 66, 94)( 67, 87)( 68, 89)( 69, 88)( 70, 90)( 71, 95)( 72, 97)( 73, 96)( 74, 98)(112,113)(115,119)(116,121)(117,120)(118,122)(123,135)(124,137)(125,136)(126,138)(127,143)(128,145)(129,144)(130,146)(131,139)(132,141)(133,140)(134,142)(147,187)(148,189)(149,188)(150,190)(151,183)(152,185)(153,184)(154,186)(155,191)(156,193)(157,192)(158,194)(159,211)(160,213)(161,212)(162,214)(163,207)(164,209)(165,208)(166,210)(167,215)(168,217)(169,216)(170,218)(171,199)(172,201)(173,200)(174,202)(175,195)(176,197)(177,196)(178,198)(179,203)(180,205)(181,204)(182,206);
s3 := Sym(218)!(  3, 87)(  4, 90)(  5, 89)(  6, 88)(  7, 95)(  8, 98)(  9, 97)( 10, 96)( 11, 91)( 12, 94)( 13, 93)( 14, 92)( 15, 75)( 16, 78)( 17, 77)( 18, 76)( 19, 83)( 20, 86)( 21, 85)( 22, 84)( 23, 79)( 24, 82)( 25, 81)( 26, 80)( 27, 99)( 28,102)( 29,101)( 30,100)( 31,107)( 32,110)( 33,109)( 34,108)( 35,103)( 36,106)( 37,105)( 38,104)( 39, 51)( 40, 54)( 41, 53)( 42, 52)( 43, 59)( 44, 62)( 45, 61)( 46, 60)( 47, 55)( 48, 58)( 49, 57)( 50, 56)( 64, 66)( 67, 71)( 68, 74)( 69, 73)( 70, 72)(111,195)(112,198)(113,197)(114,196)(115,203)(116,206)(117,205)(118,204)(119,199)(120,202)(121,201)(122,200)(123,183)(124,186)(125,185)(126,184)(127,191)(128,194)(129,193)(130,192)(131,187)(132,190)(133,189)(134,188)(135,207)(136,210)(137,209)(138,208)(139,215)(140,218)(141,217)(142,216)(143,211)(144,214)(145,213)(146,212)(147,159)(148,162)(149,161)(150,160)(151,167)(152,170)(153,169)(154,168)(155,163)(156,166)(157,165)(158,164)(172,174)(175,179)(176,182)(177,181)(178,180);
s4 := Sym(218)!( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 25, 37)( 26, 38)( 51, 63)( 52, 64)( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 61, 73)( 62, 74)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)( 93,105)( 94,106)( 95,107)( 96,108)( 97,109)( 98,110)(123,135)(124,136)(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)(133,145)(134,146)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)(165,177)(166,178)(167,179)(168,180)(169,181)(170,182)(195,207)(196,208)(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)(206,218);
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*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2*s1*s2, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s4*s3*s2*s3*s2*s3*s4*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;