Polytope of Type {20,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,12}*1440
if this polytope has a name.
Group : SmallGroup(1440,5199)
Rank : 3
Schlafli Type : {20,12}
Number of vertices, edges, etc : 60, 360, 36
Order of s0s1s2 : 20
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {20,6}*720
   4-fold quotients : {20,6}*360
   5-fold quotients : {4,12}*288
   9-fold quotients : {20,4}*160
   10-fold quotients : {4,6}*144
   18-fold quotients : {20,2}*80, {10,4}*80
   20-fold quotients : {4,6}*72
   36-fold quotients : {10,2}*40
   45-fold quotients : {4,4}*32
   72-fold quotients : {5,2}*20
   90-fold quotients : {2,4}*16, {4,2}*16
   180-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*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 2.
      18 facets:
         18 of {20}*40
      30 vertex figures:
         30 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      12 facets:
         12 of {20}*40
      20 vertex figures:
         20 of {12}*24
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2> of order 3.
      12 facets:
         12 of {20}*40
      40 vertex figures:
         30 of {4}*8
         10 of {12}*24

Permutation Representation (GAP) : 
s0 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 16, 31)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 36)( 22, 40)( 23, 39)( 24, 38)( 25, 37)( 26, 41)( 27, 45)( 28, 44)( 29, 43)( 30, 42)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)( 58, 59)( 61, 76)( 62, 80)( 63, 79)( 64, 78)( 65, 77)( 66, 81)( 67, 85)( 68, 84)( 69, 83)( 70, 82)( 71, 86)( 72, 90)( 73, 89)( 74, 88)( 75, 87)( 92, 95)( 93, 94)( 97,100)( 98, 99)(102,105)(103,104)(106,121)(107,125)(108,124)(109,123)(110,122)(111,126)(112,130)(113,129)(114,128)(115,127)(116,131)(117,135)(118,134)(119,133)(120,132)(137,140)(138,139)(142,145)(143,144)(147,150)(148,149)(151,166)(152,170)(153,169)(154,168)(155,167)(156,171)(157,175)(158,174)(159,173)(160,172)(161,176)(162,180)(163,179)(164,178)(165,177);;
s1 := (  1,  2)(  3,  5)(  6, 32)(  7, 31)(  8, 35)(  9, 34)( 10, 33)( 11, 17)( 12, 16)( 13, 20)( 14, 19)( 15, 18)( 21, 42)( 22, 41)( 23, 45)( 24, 44)( 25, 43)( 26, 27)( 28, 30)( 36, 37)( 38, 40)( 46, 47)( 48, 50)( 51, 77)( 52, 76)( 53, 80)( 54, 79)( 55, 78)( 56, 62)( 57, 61)( 58, 65)( 59, 64)( 60, 63)( 66, 87)( 67, 86)( 68, 90)( 69, 89)( 70, 88)( 71, 72)( 73, 75)( 81, 82)( 83, 85)( 91,137)( 92,136)( 93,140)( 94,139)( 95,138)( 96,167)( 97,166)( 98,170)( 99,169)(100,168)(101,152)(102,151)(103,155)(104,154)(105,153)(106,147)(107,146)(108,150)(109,149)(110,148)(111,177)(112,176)(113,180)(114,179)(115,178)(116,162)(117,161)(118,165)(119,164)(120,163)(121,142)(122,141)(123,145)(124,144)(125,143)(126,172)(127,171)(128,175)(129,174)(130,173)(131,157)(132,156)(133,160)(134,159)(135,158);;
s2 := (  1, 96)(  2, 97)(  3, 98)(  4, 99)(  5,100)(  6, 91)(  7, 92)(  8, 93)(  9, 94)( 10, 95)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)( 16,126)( 17,127)( 18,128)( 19,129)( 20,130)( 21,121)( 22,122)( 23,123)( 24,124)( 25,125)( 26,131)( 27,132)( 28,133)( 29,134)( 30,135)( 31,111)( 32,112)( 33,113)( 34,114)( 35,115)( 36,106)( 37,107)( 38,108)( 39,109)( 40,110)( 41,116)( 42,117)( 43,118)( 44,119)( 45,120)( 46,141)( 47,142)( 48,143)( 49,144)( 50,145)( 51,136)( 52,137)( 53,138)( 54,139)( 55,140)( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,171)( 62,172)( 63,173)( 64,174)( 65,175)( 66,166)( 67,167)( 68,168)( 69,169)( 70,170)( 71,176)( 72,177)( 73,178)( 74,179)( 75,180)( 76,156)( 77,157)( 78,158)( 79,159)( 80,160)( 81,151)( 82,152)( 83,153)( 84,154)( 85,155)( 86,161)( 87,162)( 88,163)( 89,164)( 90,165);;
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*s2*s1*s2*s0*s1*s2*s0*s1*s2*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, 
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(180)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 16, 31)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 36)( 22, 40)( 23, 39)( 24, 38)( 25, 37)( 26, 41)( 27, 45)( 28, 44)( 29, 43)( 30, 42)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)( 58, 59)( 61, 76)( 62, 80)( 63, 79)( 64, 78)( 65, 77)( 66, 81)( 67, 85)( 68, 84)( 69, 83)( 70, 82)( 71, 86)( 72, 90)( 73, 89)( 74, 88)( 75, 87)( 92, 95)( 93, 94)( 97,100)( 98, 99)(102,105)(103,104)(106,121)(107,125)(108,124)(109,123)(110,122)(111,126)(112,130)(113,129)(114,128)(115,127)(116,131)(117,135)(118,134)(119,133)(120,132)(137,140)(138,139)(142,145)(143,144)(147,150)(148,149)(151,166)(152,170)(153,169)(154,168)(155,167)(156,171)(157,175)(158,174)(159,173)(160,172)(161,176)(162,180)(163,179)(164,178)(165,177);
s1 := Sym(180)!(  1,  2)(  3,  5)(  6, 32)(  7, 31)(  8, 35)(  9, 34)( 10, 33)( 11, 17)( 12, 16)( 13, 20)( 14, 19)( 15, 18)( 21, 42)( 22, 41)( 23, 45)( 24, 44)( 25, 43)( 26, 27)( 28, 30)( 36, 37)( 38, 40)( 46, 47)( 48, 50)( 51, 77)( 52, 76)( 53, 80)( 54, 79)( 55, 78)( 56, 62)( 57, 61)( 58, 65)( 59, 64)( 60, 63)( 66, 87)( 67, 86)( 68, 90)( 69, 89)( 70, 88)( 71, 72)( 73, 75)( 81, 82)( 83, 85)( 91,137)( 92,136)( 93,140)( 94,139)( 95,138)( 96,167)( 97,166)( 98,170)( 99,169)(100,168)(101,152)(102,151)(103,155)(104,154)(105,153)(106,147)(107,146)(108,150)(109,149)(110,148)(111,177)(112,176)(113,180)(114,179)(115,178)(116,162)(117,161)(118,165)(119,164)(120,163)(121,142)(122,141)(123,145)(124,144)(125,143)(126,172)(127,171)(128,175)(129,174)(130,173)(131,157)(132,156)(133,160)(134,159)(135,158);
s2 := Sym(180)!(  1, 96)(  2, 97)(  3, 98)(  4, 99)(  5,100)(  6, 91)(  7, 92)(  8, 93)(  9, 94)( 10, 95)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)( 16,126)( 17,127)( 18,128)( 19,129)( 20,130)( 21,121)( 22,122)( 23,123)( 24,124)( 25,125)( 26,131)( 27,132)( 28,133)( 29,134)( 30,135)( 31,111)( 32,112)( 33,113)( 34,114)( 35,115)( 36,106)( 37,107)( 38,108)( 39,109)( 40,110)( 41,116)( 42,117)( 43,118)( 44,119)( 45,120)( 46,141)( 47,142)( 48,143)( 49,144)( 50,145)( 51,136)( 52,137)( 53,138)( 54,139)( 55,140)( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,171)( 62,172)( 63,173)( 64,174)( 65,175)( 66,166)( 67,167)( 68,168)( 69,169)( 70,170)( 71,176)( 72,177)( 73,178)( 74,179)( 75,180)( 76,156)( 77,157)( 78,158)( 79,159)( 80,160)( 81,151)( 82,152)( 83,153)( 84,154)( 85,155)( 86,161)( 87,162)( 88,163)( 89,164)( 90,165);
poly := sub<Sym(180)|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*s2*s1*s2*s0*s1*s2*s0*s1*s2*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, 
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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