Part of the Atlas of Small Regular Polytopes

Polytope of Type {18,4}

Atlas Canonical Name {18,4}*1944c

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1944,806)
Rank
3
Schläfli Type
{18,4}
Vertices, edges, …
243, 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

81 vertex figures

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

18 facets

81 vertex figures

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

18 facets

81 vertex figures

P/N, where N=<s0*(s1*s0*s2)^3*s1*s2, (s0*s1)^2*s0*s2*(s1*s0)^2*s1*s2*s1> of order 9

12 facets

27 vertex figures

P/N, where N=<(s0*s1)^2*s2*(s1*s0)^3*s2*s1, s0*s1*s2*(s1*s0)^3*s2*s1*s0*s1> of order 9

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

References

None.

to this polytope.

Twisty Puzzle