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}*1440
if this polytope has a name.
Group : SmallGroup(1440,5199)
Rank : 3
Schlafli Type : {12,20}
Number of vertices, edges, etc : 36, 360, 60
Order of s0s1s2 : 20
Order of s0s1s2s1 : 6
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 : {6,20}*720
   4-fold quotients : {6,20}*360
   5-fold quotients : {12,4}*288
   9-fold quotients : {4,20}*160
   10-fold quotients : {6,4}*144
   18-fold quotients : {2,20}*80, {4,10}*80
   20-fold quotients : {6,4}*72
   36-fold quotients : {2,10}*40
   45-fold quotients : {4,4}*32
   72-fold quotients : {2,5}*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*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      30 facets:
         30 of {12}*24
      18 vertex figures:
         18 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      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> of order 3.
      40 facets:
         30 of {4}*8
         10 of {12}*24
      12 vertex figures:
         12 of {20}*40

Permutation Representation (GAP) : 
s0 := (  1, 91)(  2, 92)(  3, 93)(  4, 94)(  5, 95)(  6,101)(  7,102)(  8,103)(  9,104)( 10,105)( 11, 96)( 12, 97)( 13, 98)( 14, 99)( 15,100)( 16,121)( 17,122)( 18,123)( 19,124)( 20,125)( 21,131)( 22,132)( 23,133)( 24,134)( 25,135)( 26,126)( 27,127)( 28,128)( 29,129)( 30,130)( 31,106)( 32,107)( 33,108)( 34,109)( 35,110)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)( 41,111)( 42,112)( 43,113)( 44,114)( 45,115)( 46,136)( 47,137)( 48,138)( 49,139)( 50,140)( 51,146)( 52,147)( 53,148)( 54,149)( 55,150)( 56,141)( 57,142)( 58,143)( 59,144)( 60,145)( 61,166)( 62,167)( 63,168)( 64,169)( 65,170)( 66,176)( 67,177)( 68,178)( 69,179)( 70,180)( 71,171)( 72,172)( 73,173)( 74,174)( 75,175)( 76,151)( 77,152)( 78,153)( 79,154)( 80,155)( 81,161)( 82,162)( 83,163)( 84,164)( 85,165)( 86,156)( 87,157)( 88,158)( 89,159)( 90,160);;
s1 := (  1, 21)(  2, 25)(  3, 24)(  4, 23)(  5, 22)(  7, 10)(  8,  9)( 11, 36)( 12, 40)( 13, 39)( 14, 38)( 15, 37)( 17, 20)( 18, 19)( 26, 31)( 27, 35)( 28, 34)( 29, 33)( 30, 32)( 42, 45)( 43, 44)( 46, 66)( 47, 70)( 48, 69)( 49, 68)( 50, 67)( 52, 55)( 53, 54)( 56, 81)( 57, 85)( 58, 84)( 59, 83)( 60, 82)( 62, 65)( 63, 64)( 71, 76)( 72, 80)( 73, 79)( 74, 78)( 75, 77)( 87, 90)( 88, 89)( 91,156)( 92,160)( 93,159)( 94,158)( 95,157)( 96,141)( 97,145)( 98,144)( 99,143)(100,142)(101,171)(102,175)(103,174)(104,173)(105,172)(106,151)(107,155)(108,154)(109,153)(110,152)(111,136)(112,140)(113,139)(114,138)(115,137)(116,166)(117,170)(118,169)(119,168)(120,167)(121,161)(122,165)(123,164)(124,163)(125,162)(126,146)(127,150)(128,149)(129,148)(130,147)(131,176)(132,180)(133,179)(134,178)(135,177);;
s2 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 32)( 17, 31)( 18, 35)( 19, 34)( 20, 33)( 21, 37)( 22, 36)( 23, 40)( 24, 39)( 25, 38)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)( 58, 60)( 61, 77)( 62, 76)( 63, 80)( 64, 79)( 65, 78)( 66, 82)( 67, 81)( 68, 85)( 69, 84)( 70, 83)( 71, 87)( 72, 86)( 73, 90)( 74, 89)( 75, 88)( 91, 92)( 93, 95)( 96, 97)( 98,100)(101,102)(103,105)(106,122)(107,121)(108,125)(109,124)(110,123)(111,127)(112,126)(113,130)(114,129)(115,128)(116,132)(117,131)(118,135)(119,134)(120,133)(136,137)(138,140)(141,142)(143,145)(146,147)(148,150)(151,167)(152,166)(153,170)(154,169)(155,168)(156,172)(157,171)(158,175)(159,174)(160,173)(161,177)(162,176)(163,180)(164,179)(165,178);;
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*s1*s0*s1*s0*s2*s1*s0*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*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*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(180)!(  1, 91)(  2, 92)(  3, 93)(  4, 94)(  5, 95)(  6,101)(  7,102)(  8,103)(  9,104)( 10,105)( 11, 96)( 12, 97)( 13, 98)( 14, 99)( 15,100)( 16,121)( 17,122)( 18,123)( 19,124)( 20,125)( 21,131)( 22,132)( 23,133)( 24,134)( 25,135)( 26,126)( 27,127)( 28,128)( 29,129)( 30,130)( 31,106)( 32,107)( 33,108)( 34,109)( 35,110)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)( 41,111)( 42,112)( 43,113)( 44,114)( 45,115)( 46,136)( 47,137)( 48,138)( 49,139)( 50,140)( 51,146)( 52,147)( 53,148)( 54,149)( 55,150)( 56,141)( 57,142)( 58,143)( 59,144)( 60,145)( 61,166)( 62,167)( 63,168)( 64,169)( 65,170)( 66,176)( 67,177)( 68,178)( 69,179)( 70,180)( 71,171)( 72,172)( 73,173)( 74,174)( 75,175)( 76,151)( 77,152)( 78,153)( 79,154)( 80,155)( 81,161)( 82,162)( 83,163)( 84,164)( 85,165)( 86,156)( 87,157)( 88,158)( 89,159)( 90,160);
s1 := Sym(180)!(  1, 21)(  2, 25)(  3, 24)(  4, 23)(  5, 22)(  7, 10)(  8,  9)( 11, 36)( 12, 40)( 13, 39)( 14, 38)( 15, 37)( 17, 20)( 18, 19)( 26, 31)( 27, 35)( 28, 34)( 29, 33)( 30, 32)( 42, 45)( 43, 44)( 46, 66)( 47, 70)( 48, 69)( 49, 68)( 50, 67)( 52, 55)( 53, 54)( 56, 81)( 57, 85)( 58, 84)( 59, 83)( 60, 82)( 62, 65)( 63, 64)( 71, 76)( 72, 80)( 73, 79)( 74, 78)( 75, 77)( 87, 90)( 88, 89)( 91,156)( 92,160)( 93,159)( 94,158)( 95,157)( 96,141)( 97,145)( 98,144)( 99,143)(100,142)(101,171)(102,175)(103,174)(104,173)(105,172)(106,151)(107,155)(108,154)(109,153)(110,152)(111,136)(112,140)(113,139)(114,138)(115,137)(116,166)(117,170)(118,169)(119,168)(120,167)(121,161)(122,165)(123,164)(124,163)(125,162)(126,146)(127,150)(128,149)(129,148)(130,147)(131,176)(132,180)(133,179)(134,178)(135,177);
s2 := Sym(180)!(  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 32)( 17, 31)( 18, 35)( 19, 34)( 20, 33)( 21, 37)( 22, 36)( 23, 40)( 24, 39)( 25, 38)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)( 58, 60)( 61, 77)( 62, 76)( 63, 80)( 64, 79)( 65, 78)( 66, 82)( 67, 81)( 68, 85)( 69, 84)( 70, 83)( 71, 87)( 72, 86)( 73, 90)( 74, 89)( 75, 88)( 91, 92)( 93, 95)( 96, 97)( 98,100)(101,102)(103,105)(106,122)(107,121)(108,125)(109,124)(110,123)(111,127)(112,126)(113,130)(114,129)(115,128)(116,132)(117,131)(118,135)(119,134)(120,133)(136,137)(138,140)(141,142)(143,145)(146,147)(148,150)(151,167)(152,166)(153,170)(154,169)(155,168)(156,172)(157,171)(158,175)(159,174)(160,173)(161,177)(162,176)(163,180)(164,179)(165,178);
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*s1*s0*s1*s0*s2*s1*s0*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*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*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.
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