Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,36,12}

Atlas Canonical Name {2,36,12}*1728b

Overview

Group
SmallGroup(1728,16615)
Rank
4
Schläfli Type
{2,36,12}
Vertices, edges, …
2, 36, 216, 12
Order of s0s1s2s3
36
Order of s0s1s2s3s2s1
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

8-fold

9-fold

12-fold

18-fold

24-fold

27-fold

36-fold

54-fold

72-fold

108-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)(111,192)(112,194)(113,193)(114,198)(115,200)(116,199)(117,195)(118,197)(119,196)(120,212)(121,211)(122,210)(123,218)(124,217)(125,216)(126,215)(127,214)(128,213)(129,203)(130,202)(131,201)(132,209)(133,208)(134,207)(135,206)(136,205)(137,204)(138,165)(139,167)(140,166)(141,171)(142,173)(143,172)(144,168)(145,170)(146,169)(147,185)(148,184)(149,183)(150,191)(151,190)(152,189)(153,188)(154,187)(155,186)(156,176)(157,175)(158,174)(159,182)(160,181)(161,180)(162,179)(163,178)(164,177);;
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)( 60, 63)( 61, 64)( 62, 65)( 69, 72)( 70, 73)( 71, 74)( 78, 81)( 79, 82)( 80, 83)( 87, 90)( 88, 91)( 89, 92)( 96, 99)( 97,100)( 98,101)(105,108)(106,109)(107,110)(111,138)(112,139)(113,140)(114,144)(115,145)(116,146)(117,141)(118,142)(119,143)(120,147)(121,148)(122,149)(123,153)(124,154)(125,155)(126,150)(127,151)(128,152)(129,156)(130,157)(131,158)(132,162)(133,163)(134,164)(135,159)(136,160)(137,161)(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);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*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*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)(111,192)(112,194)(113,193)(114,198)(115,200)(116,199)(117,195)(118,197)(119,196)(120,212)(121,211)(122,210)(123,218)(124,217)(125,216)(126,215)(127,214)(128,213)(129,203)(130,202)(131,201)(132,209)(133,208)(134,207)(135,206)(136,205)(137,204)(138,165)(139,167)(140,166)(141,171)(142,173)(143,172)(144,168)(145,170)(146,169)(147,185)(148,184)(149,183)(150,191)(151,190)(152,189)(153,188)(154,187)(155,186)(156,176)(157,175)(158,174)(159,182)(160,181)(161,180)(162,179)(163,178)(164,177);
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)( 60, 63)( 61, 64)( 62, 65)( 69, 72)( 70, 73)( 71, 74)( 78, 81)( 79, 82)( 80, 83)( 87, 90)( 88, 91)( 89, 92)( 96, 99)( 97,100)( 98,101)(105,108)(106,109)(107,110)(111,138)(112,139)(113,140)(114,144)(115,145)(116,146)(117,141)(118,142)(119,143)(120,147)(121,148)(122,149)(123,153)(124,154)(125,155)(126,150)(127,151)(128,152)(129,156)(130,157)(131,158)(132,162)(133,163)(134,164)(135,159)(136,160)(137,161)(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);
poly := sub<Sym(218)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*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*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 >;