Part of the Atlas of Small Regular Polytopes

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

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

Overview

Group
SmallGroup(1728,46114)
Rank
5
Schläfli Type
{4,6,18,2}
Vertices, edges, …
4, 12, 54, 18, 2
Order of s0s1s2s3s4
18
Order of s0s1s2s3s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

18-fold

Covers minimal covers in bold

None in this atlas.

Representations

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