Part of the Atlas of Small Regular Polytopes

Polytope of Type {18,12}

Atlas Canonical Name {18,12}*1944e

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Overview

Group
SmallGroup(1944,806)
Rank
3
Schläfli Type
{18,12}
Vertices, edges, …
81, 486, 54
Order of s0s1s2
12
Order of s0s1s2s1
18
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable

Quotients maximal quotients in bold

9-fold

27-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s0*s2*(s1*s0)^5*s1*s2> of order 3

36 facets

27 vertex figures

P/N, where N=<(s0*s1)^2*s2*(s1*s0)^3*s2*s1> of order 3

18 facets

27 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 17)( 11, 16)( 12, 18)( 13, 14)( 19, 24)( 20, 23)( 21, 22)( 25, 27)( 28, 75)( 29, 74)( 30, 73)( 31, 81)( 32, 80)( 33, 79)( 34, 78)( 35, 77)( 36, 76)( 37, 61)( 38, 63)( 39, 62)( 40, 58)( 41, 60)( 42, 59)( 43, 55)( 44, 57)( 45, 56)( 46, 68)( 47, 67)( 48, 69)( 49, 65)( 50, 64)( 51, 66)( 52, 71)( 53, 70)( 54, 72)( 82,168)( 83,167)( 84,166)( 85,165)( 86,164)( 87,163)( 88,171)( 89,170)( 90,169)( 91,172)( 92,174)( 93,173)( 94,178)( 95,180)( 96,179)( 97,175)( 98,177)( 99,176)(100,188)(101,187)(102,189)(103,185)(104,184)(105,186)(106,182)(107,181)(108,183)(109,239)(110,238)(111,240)(112,236)(113,235)(114,237)(115,242)(116,241)(117,243)(118,219)(119,218)(120,217)(121,225)(122,224)(123,223)(124,222)(125,221)(126,220)(127,232)(128,234)(129,233)(130,229)(131,231)(132,230)(133,226)(134,228)(135,227)(136,201)(137,200)(138,199)(139,207)(140,206)(141,205)(142,204)(143,203)(144,202)(145,214)(146,216)(147,215)(148,211)(149,213)(150,212)(151,208)(152,210)(153,209)(154,194)(155,193)(156,195)(157,191)(158,190)(159,192)(160,197)(161,196)(162,198);;
s1 := (  1, 82)(  2, 90)(  3, 86)(  4, 85)(  5, 84)(  6, 89)(  7, 88)(  8, 87)(  9, 83)( 10,106)( 11,105)( 12,101)( 13,100)( 14,108)( 15,104)( 16,103)( 17,102)( 18,107)( 19, 94)( 20, 93)( 21, 98)( 22, 97)( 23, 96)( 24, 92)( 25, 91)( 26, 99)( 27, 95)( 28,135)( 29,131)( 30,127)( 31,129)( 32,134)( 33,130)( 34,132)( 35,128)( 36,133)( 37,123)( 38,119)( 39,124)( 40,126)( 41,122)( 42,118)( 43,120)( 44,125)( 45,121)( 46,111)( 47,116)( 48,112)( 49,114)( 50,110)( 51,115)( 52,117)( 53,113)( 54,109)( 55,147)( 56,152)( 57,148)( 58,150)( 59,146)( 60,151)( 61,153)( 62,149)( 63,145)( 64,144)( 65,140)( 66,136)( 67,138)( 68,143)( 69,139)( 70,141)( 71,137)( 72,142)( 73,159)( 74,155)( 75,160)( 76,162)( 77,158)( 78,154)( 79,156)( 80,161)( 81,157)(163,168)(165,169)(166,171)(172,183)(173,188)(174,184)(175,186)(176,182)(177,187)(178,189)(179,185)(180,181)(190,209)(191,214)(192,213)(193,212)(194,208)(195,216)(196,215)(197,211)(198,210)(199,206)(200,202)(203,205)(217,230)(218,226)(219,234)(220,233)(221,229)(222,228)(223,227)(224,232)(225,231)(235,242)(236,238)(239,241);;
s2 := (  2,  3)(  4,  6)(  7,  8)( 10, 23)( 11, 22)( 12, 24)( 13, 25)( 14, 27)( 15, 26)( 16, 21)( 17, 20)( 18, 19)( 28,181)( 29,183)( 30,182)( 31,186)( 32,185)( 33,184)( 34,188)( 35,187)( 36,189)( 37,176)( 38,175)( 39,177)( 40,178)( 41,180)( 42,179)( 43,174)( 44,173)( 45,172)( 46,171)( 47,170)( 48,169)( 49,164)( 50,163)( 51,165)( 52,166)( 53,168)( 54,167)( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61, 99)( 62, 98)( 63, 97)( 64, 87)( 65, 86)( 66, 85)( 67, 89)( 68, 88)( 69, 90)( 70, 82)( 71, 84)( 72, 83)( 73,106)( 74,108)( 75,107)( 76,102)( 77,101)( 78,100)( 79,104)( 80,103)( 81,105)(109,233)(110,232)(111,234)(112,226)(113,228)(114,227)(115,231)(116,230)(117,229)(118,219)(119,218)(120,217)(121,221)(122,220)(123,222)(124,223)(125,225)(126,224)(127,238)(128,240)(129,239)(130,243)(131,242)(132,241)(133,236)(134,235)(135,237)(136,148)(137,150)(138,149)(139,153)(140,152)(141,151)(142,146)(143,145)(144,147)(154,156)(157,158)(161,162)(190,191)(194,195)(196,198)(199,213)(200,212)(201,211)(202,215)(203,214)(204,216)(205,208)(206,210)(207,209);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(243)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 17)( 11, 16)( 12, 18)( 13, 14)( 19, 24)( 20, 23)( 21, 22)( 25, 27)( 28, 75)( 29, 74)( 30, 73)( 31, 81)( 32, 80)( 33, 79)( 34, 78)( 35, 77)( 36, 76)( 37, 61)( 38, 63)( 39, 62)( 40, 58)( 41, 60)( 42, 59)( 43, 55)( 44, 57)( 45, 56)( 46, 68)( 47, 67)( 48, 69)( 49, 65)( 50, 64)( 51, 66)( 52, 71)( 53, 70)( 54, 72)( 82,168)( 83,167)( 84,166)( 85,165)( 86,164)( 87,163)( 88,171)( 89,170)( 90,169)( 91,172)( 92,174)( 93,173)( 94,178)( 95,180)( 96,179)( 97,175)( 98,177)( 99,176)(100,188)(101,187)(102,189)(103,185)(104,184)(105,186)(106,182)(107,181)(108,183)(109,239)(110,238)(111,240)(112,236)(113,235)(114,237)(115,242)(116,241)(117,243)(118,219)(119,218)(120,217)(121,225)(122,224)(123,223)(124,222)(125,221)(126,220)(127,232)(128,234)(129,233)(130,229)(131,231)(132,230)(133,226)(134,228)(135,227)(136,201)(137,200)(138,199)(139,207)(140,206)(141,205)(142,204)(143,203)(144,202)(145,214)(146,216)(147,215)(148,211)(149,213)(150,212)(151,208)(152,210)(153,209)(154,194)(155,193)(156,195)(157,191)(158,190)(159,192)(160,197)(161,196)(162,198);
s1 := Sym(243)!(  1, 82)(  2, 90)(  3, 86)(  4, 85)(  5, 84)(  6, 89)(  7, 88)(  8, 87)(  9, 83)( 10,106)( 11,105)( 12,101)( 13,100)( 14,108)( 15,104)( 16,103)( 17,102)( 18,107)( 19, 94)( 20, 93)( 21, 98)( 22, 97)( 23, 96)( 24, 92)( 25, 91)( 26, 99)( 27, 95)( 28,135)( 29,131)( 30,127)( 31,129)( 32,134)( 33,130)( 34,132)( 35,128)( 36,133)( 37,123)( 38,119)( 39,124)( 40,126)( 41,122)( 42,118)( 43,120)( 44,125)( 45,121)( 46,111)( 47,116)( 48,112)( 49,114)( 50,110)( 51,115)( 52,117)( 53,113)( 54,109)( 55,147)( 56,152)( 57,148)( 58,150)( 59,146)( 60,151)( 61,153)( 62,149)( 63,145)( 64,144)( 65,140)( 66,136)( 67,138)( 68,143)( 69,139)( 70,141)( 71,137)( 72,142)( 73,159)( 74,155)( 75,160)( 76,162)( 77,158)( 78,154)( 79,156)( 80,161)( 81,157)(163,168)(165,169)(166,171)(172,183)(173,188)(174,184)(175,186)(176,182)(177,187)(178,189)(179,185)(180,181)(190,209)(191,214)(192,213)(193,212)(194,208)(195,216)(196,215)(197,211)(198,210)(199,206)(200,202)(203,205)(217,230)(218,226)(219,234)(220,233)(221,229)(222,228)(223,227)(224,232)(225,231)(235,242)(236,238)(239,241);
s2 := Sym(243)!(  2,  3)(  4,  6)(  7,  8)( 10, 23)( 11, 22)( 12, 24)( 13, 25)( 14, 27)( 15, 26)( 16, 21)( 17, 20)( 18, 19)( 28,181)( 29,183)( 30,182)( 31,186)( 32,185)( 33,184)( 34,188)( 35,187)( 36,189)( 37,176)( 38,175)( 39,177)( 40,178)( 41,180)( 42,179)( 43,174)( 44,173)( 45,172)( 46,171)( 47,170)( 48,169)( 49,164)( 50,163)( 51,165)( 52,166)( 53,168)( 54,167)( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61, 99)( 62, 98)( 63, 97)( 64, 87)( 65, 86)( 66, 85)( 67, 89)( 68, 88)( 69, 90)( 70, 82)( 71, 84)( 72, 83)( 73,106)( 74,108)( 75,107)( 76,102)( 77,101)( 78,100)( 79,104)( 80,103)( 81,105)(109,233)(110,232)(111,234)(112,226)(113,228)(114,227)(115,231)(116,230)(117,229)(118,219)(119,218)(120,217)(121,221)(122,220)(123,222)(124,223)(125,225)(126,224)(127,238)(128,240)(129,239)(130,243)(131,242)(132,241)(133,236)(134,235)(135,237)(136,148)(137,150)(138,149)(139,153)(140,152)(141,151)(142,146)(143,145)(144,147)(154,156)(157,158)(161,162)(190,191)(194,195)(196,198)(199,213)(200,212)(201,211)(202,215)(203,214)(204,216)(205,208)(206,210)(207,209);
poly := sub<Sym(243)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1 >;

References

None.

to this polytope.

Twisty Puzzle