Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,18,6,4}

Atlas Canonical Name {2,18,6,4}*1728b

Overview

Group
SmallGroup(1728,30872)
Rank
5
Schläfli Type
{2,18,6,4}
Vertices, edges, …
2, 18, 54, 12, 4
Order of s0s1s2s3s4
36
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

27-fold

36-fold

54-fold

Covers minimal covers in bold

None in this atlas.

Representations

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