Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,18}

Atlas Canonical Name {3,18}*972a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(972,100)
Rank
3
Schläfli Type
{3,18}
Vertices, edges, …
27, 243, 162
Order of s0s1s2
18
Order of s0s1s2s1
18
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

9-fold

27-fold

81-fold

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

54 facets

9 vertex figures

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

54 facets

9 vertex figures

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

18 facets

3 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle