Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,30}

Atlas Canonical Name {12,30}*960b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(960,10994)
Rank
3
Schläfli Type
{12,30}
Vertices, edges, …
16, 240, 40
Order of s0s1s2
20
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

4-fold

5-fold

10-fold

12-fold

20-fold

24-fold

40-fold

48-fold

60-fold

120-fold

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

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

References

None.

to this polytope.

Twisty Puzzle