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}*1728l
if this polytope has a name.
Group : SmallGroup(1728,12713)
Rank : 3
Schlafli Type : {24,12}
Number of vertices, edges, etc : 72, 432, 36
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
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}*864g
   3-fold quotients : {24,4}*576b
   4-fold quotients : {6,12}*432e
   6-fold quotients : {12,4}*288
   8-fold quotients : {6,12}*216a
   12-fold quotients : {6,4}*144
   24-fold quotients : {6,4}*72
   27-fold quotients : {8,4}*64b
   54-fold quotients : {4,4}*32
   108-fold quotients : {2,4}*16, {4,2}*16
   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=<s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 3.
      12 facets:
         12 of {24}*48
      24 vertex figures:
         24 of {12}*24

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