Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6}

Atlas Canonical Name {12,6}*1944a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1944,805)
Rank
3
Schläfli Type
{12,6}
Vertices, edges, …
162, 486, 81
Order of s0s1s2
4
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*s1*s0*(s2*(s1*s0)^2)^2*s2*s1> of order 3

27 facets

54 vertex figures

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

27 facets

54 vertex figures

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

33 facets

54 vertex figures

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

15 facets

18 vertex figures

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

15 facets

18 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle