Polytope of Type {24,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,12}*1728v
if this polytope has a name.
Group : SmallGroup(1728,33596)
Rank : 3
Schlafli Type : {24,12}
Number of vertices, edges, etc : 72, 432, 36
Order of s0s1s2 : 24
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 : {12,12}*864l
   3-fold quotients : {24,4}*576a
   4-fold quotients : {6,12}*432i
   6-fold quotients : {12,4}*288
   8-fold quotients : {6,12}*216c
   9-fold quotients : {8,12}*192a
   12-fold quotients : {6,4}*144
   18-fold quotients : {4,12}*96a, {8,6}*96
   24-fold quotients : {6,4}*72
   27-fold quotients : {8,4}*64a
   36-fold quotients : {2,12}*48, {4,6}*48a
   54-fold quotients : {4,4}*32, {8,2}*32
   72-fold quotients : {2,6}*24
   108-fold quotients : {2,4}*16, {4,2}*16
   144-fold quotients : {2,3}*12
   216-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> of order 2.
      18 facets:
         18 of {24}*48
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
      12 facets:
         12 of {24}*48
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      12 facets:
         12 of {24}*48
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 3.
      24 facets:
         18 of {8}*16
         6 of {24}*48
      24 vertex figures:
         24 of {12}*24

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