Polytope of Type {12,20}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,20}*1920a
if this polytope has a name.
Group : SmallGroup(1920,42362)
Rank : 3
Schlafli Type : {12,20}
Number of vertices, edges, etc : 48, 480, 80
Order of s0s1s2 : 120
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,20}*960a
   3-fold quotients : {4,20}*640a
   4-fold quotients : {12,20}*480
   5-fold quotients : {12,4}*384a
   6-fold quotients : {4,20}*320
   8-fold quotients : {12,10}*240, {6,20}*240a
   10-fold quotients : {12,4}*192a
   12-fold quotients : {4,20}*160
   15-fold quotients : {4,4}*128
   16-fold quotients : {6,10}*120
   20-fold quotients : {12,4}*96a
   24-fold quotients : {2,20}*80, {4,10}*80
   30-fold quotients : {4,4}*64
   40-fold quotients : {12,2}*48, {6,4}*48a
   48-fold quotients : {2,10}*40
   60-fold quotients : {4,4}*32
   80-fold quotients : {6,2}*24
   96-fold quotients : {2,5}*20
   120-fold quotients : {2,4}*16, {4,2}*16
   160-fold quotients : {3,2}*12
   240-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      50 facets:
         20 of {6}*12
         30 of {12}*24
      24 vertex figures:
         24 of {20}*40
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
      40 facets:
         40 of {12}*24
      24 vertex figures:
         24 of {20}*40
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      40 facets:
         40 of {12}*24
      24 vertex figures:
         24 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      40 facets:
         40 of {12}*24
      24 vertex figures:
         24 of {20}*40
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      40 facets:
         40 of {12}*24
      30 vertex figures:
         12 of {10}*20
         18 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0> of order 4.
      20 facets:
         20 of {12}*24
      15 vertex figures:
         6 of {10}*20
         9 of {20}*40
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 4.
      30 facets:
         20 of {6}*12
         10 of {12}*24
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      20 facets:
         20 of {12}*24
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      20 facets:
         20 of {12}*24
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
      25 facets:
         10 of {6}*12
         15 of {12}*24
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      20 facets:
         20 of {12}*24
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      20 facets:
         20 of {12}*24
      18 vertex figures:
         12 of {10}*20
         6 of {20}*40

Permutation Representation (GAP) : 
s0 := (  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 21, 26)( 22, 27)( 23, 28)( 24, 29)( 25, 30)( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 61,106)( 62,107)( 63,108)( 64,109)( 65,110)( 66,116)( 67,117)( 68,118)( 69,119)( 70,120)( 71,111)( 72,112)( 73,113)( 74,114)( 75,115)( 76, 91)( 77, 92)( 78, 93)( 79, 94)( 80, 95)( 81,101)( 82,102)( 83,103)( 84,104)( 85,105)( 86, 96)( 87, 97)( 88, 98)( 89, 99)( 90,100)(126,131)(127,132)(128,133)(129,134)(130,135)(141,146)(142,147)(143,148)(144,149)(145,150)(156,161)(157,162)(158,163)(159,164)(160,165)(171,176)(172,177)(173,178)(174,179)(175,180)(181,226)(182,227)(183,228)(184,229)(185,230)(186,236)(187,237)(188,238)(189,239)(190,240)(191,231)(192,232)(193,233)(194,234)(195,235)(196,211)(197,212)(198,213)(199,214)(200,215)(201,221)(202,222)(203,223)(204,224)(205,225)(206,216)(207,217)(208,218)(209,219)(210,220);;
s1 := (  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 12, 15)( 13, 14)( 16, 21)( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 30)( 28, 29)( 31, 51)( 32, 55)( 33, 54)( 34, 53)( 35, 52)( 36, 46)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 41, 56)( 42, 60)( 43, 59)( 44, 58)( 45, 57)( 61, 66)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 72, 75)( 73, 74)( 76, 81)( 77, 85)( 78, 84)( 79, 83)( 80, 82)( 87, 90)( 88, 89)( 91,111)( 92,115)( 93,114)( 94,113)( 95,112)( 96,106)( 97,110)( 98,109)( 99,108)(100,107)(101,116)(102,120)(103,119)(104,118)(105,117)(121,186)(122,190)(123,189)(124,188)(125,187)(126,181)(127,185)(128,184)(129,183)(130,182)(131,191)(132,195)(133,194)(134,193)(135,192)(136,201)(137,205)(138,204)(139,203)(140,202)(141,196)(142,200)(143,199)(144,198)(145,197)(146,206)(147,210)(148,209)(149,208)(150,207)(151,231)(152,235)(153,234)(154,233)(155,232)(156,226)(157,230)(158,229)(159,228)(160,227)(161,236)(162,240)(163,239)(164,238)(165,237)(166,216)(167,220)(168,219)(169,218)(170,217)(171,211)(172,215)(173,214)(174,213)(175,212)(176,221)(177,225)(178,224)(179,223)(180,222);;
s2 := (  1,122)(  2,121)(  3,125)(  4,124)(  5,123)(  6,127)(  7,126)(  8,130)(  9,129)( 10,128)( 11,132)( 12,131)( 13,135)( 14,134)( 15,133)( 16,137)( 17,136)( 18,140)( 19,139)( 20,138)( 21,142)( 22,141)( 23,145)( 24,144)( 25,143)( 26,147)( 27,146)( 28,150)( 29,149)( 30,148)( 31,152)( 32,151)( 33,155)( 34,154)( 35,153)( 36,157)( 37,156)( 38,160)( 39,159)( 40,158)( 41,162)( 42,161)( 43,165)( 44,164)( 45,163)( 46,167)( 47,166)( 48,170)( 49,169)( 50,168)( 51,172)( 52,171)( 53,175)( 54,174)( 55,173)( 56,177)( 57,176)( 58,180)( 59,179)( 60,178)( 61,182)( 62,181)( 63,185)( 64,184)( 65,183)( 66,187)( 67,186)( 68,190)( 69,189)( 70,188)( 71,192)( 72,191)( 73,195)( 74,194)( 75,193)( 76,197)( 77,196)( 78,200)( 79,199)( 80,198)( 81,202)( 82,201)( 83,205)( 84,204)( 85,203)( 86,207)( 87,206)( 88,210)( 89,209)( 90,208)( 91,212)( 92,211)( 93,215)( 94,214)( 95,213)( 96,217)( 97,216)( 98,220)( 99,219)(100,218)(101,222)(102,221)(103,225)(104,224)(105,223)(106,227)(107,226)(108,230)(109,229)(110,228)(111,232)(112,231)(113,235)(114,234)(115,233)(116,237)(117,236)(118,240)(119,239)(120,238);;
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, s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(240)!(  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 21, 26)( 22, 27)( 23, 28)( 24, 29)( 25, 30)( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 61,106)( 62,107)( 63,108)( 64,109)( 65,110)( 66,116)( 67,117)( 68,118)( 69,119)( 70,120)( 71,111)( 72,112)( 73,113)( 74,114)( 75,115)( 76, 91)( 77, 92)( 78, 93)( 79, 94)( 80, 95)( 81,101)( 82,102)( 83,103)( 84,104)( 85,105)( 86, 96)( 87, 97)( 88, 98)( 89, 99)( 90,100)(126,131)(127,132)(128,133)(129,134)(130,135)(141,146)(142,147)(143,148)(144,149)(145,150)(156,161)(157,162)(158,163)(159,164)(160,165)(171,176)(172,177)(173,178)(174,179)(175,180)(181,226)(182,227)(183,228)(184,229)(185,230)(186,236)(187,237)(188,238)(189,239)(190,240)(191,231)(192,232)(193,233)(194,234)(195,235)(196,211)(197,212)(198,213)(199,214)(200,215)(201,221)(202,222)(203,223)(204,224)(205,225)(206,216)(207,217)(208,218)(209,219)(210,220);
s1 := Sym(240)!(  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 12, 15)( 13, 14)( 16, 21)( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 30)( 28, 29)( 31, 51)( 32, 55)( 33, 54)( 34, 53)( 35, 52)( 36, 46)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 41, 56)( 42, 60)( 43, 59)( 44, 58)( 45, 57)( 61, 66)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 72, 75)( 73, 74)( 76, 81)( 77, 85)( 78, 84)( 79, 83)( 80, 82)( 87, 90)( 88, 89)( 91,111)( 92,115)( 93,114)( 94,113)( 95,112)( 96,106)( 97,110)( 98,109)( 99,108)(100,107)(101,116)(102,120)(103,119)(104,118)(105,117)(121,186)(122,190)(123,189)(124,188)(125,187)(126,181)(127,185)(128,184)(129,183)(130,182)(131,191)(132,195)(133,194)(134,193)(135,192)(136,201)(137,205)(138,204)(139,203)(140,202)(141,196)(142,200)(143,199)(144,198)(145,197)(146,206)(147,210)(148,209)(149,208)(150,207)(151,231)(152,235)(153,234)(154,233)(155,232)(156,226)(157,230)(158,229)(159,228)(160,227)(161,236)(162,240)(163,239)(164,238)(165,237)(166,216)(167,220)(168,219)(169,218)(170,217)(171,211)(172,215)(173,214)(174,213)(175,212)(176,221)(177,225)(178,224)(179,223)(180,222);
s2 := Sym(240)!(  1,122)(  2,121)(  3,125)(  4,124)(  5,123)(  6,127)(  7,126)(  8,130)(  9,129)( 10,128)( 11,132)( 12,131)( 13,135)( 14,134)( 15,133)( 16,137)( 17,136)( 18,140)( 19,139)( 20,138)( 21,142)( 22,141)( 23,145)( 24,144)( 25,143)( 26,147)( 27,146)( 28,150)( 29,149)( 30,148)( 31,152)( 32,151)( 33,155)( 34,154)( 35,153)( 36,157)( 37,156)( 38,160)( 39,159)( 40,158)( 41,162)( 42,161)( 43,165)( 44,164)( 45,163)( 46,167)( 47,166)( 48,170)( 49,169)( 50,168)( 51,172)( 52,171)( 53,175)( 54,174)( 55,173)( 56,177)( 57,176)( 58,180)( 59,179)( 60,178)( 61,182)( 62,181)( 63,185)( 64,184)( 65,183)( 66,187)( 67,186)( 68,190)( 69,189)( 70,188)( 71,192)( 72,191)( 73,195)( 74,194)( 75,193)( 76,197)( 77,196)( 78,200)( 79,199)( 80,198)( 81,202)( 82,201)( 83,205)( 84,204)( 85,203)( 86,207)( 87,206)( 88,210)( 89,209)( 90,208)( 91,212)( 92,211)( 93,215)( 94,214)( 95,213)( 96,217)( 97,216)( 98,220)( 99,219)(100,218)(101,222)(102,221)(103,225)(104,224)(105,223)(106,227)(107,226)(108,230)(109,229)(110,228)(111,232)(112,231)(113,235)(114,234)(115,233)(116,237)(117,236)(118,240)(119,239)(120,238);
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*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope

Twisty Puzzle