Polytope of Type {24,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,6}*864b
if this polytope has a name.
Group : SmallGroup(864,814)
Rank : 3
Schlafli Type : {24,6}
Number of vertices, edges, etc : 72, 216, 18
Order of s0s1s2 : 24
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {24,6,2} of size 1728
Vertex Figure Of :
   {2,24,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,6}*432b
   3-fold quotients : {24,6}*288a
   4-fold quotients : {6,6}*216b
   6-fold quotients : {12,6}*144a
   8-fold quotients : {6,6}*108
   9-fold quotients : {24,2}*96, {8,6}*96
   12-fold quotients : {6,6}*72a
   18-fold quotients : {12,2}*48, {4,6}*48a
   27-fold quotients : {8,2}*32
   36-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {4,2}*16
   72-fold quotients : {2,3}*12, {3,2}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {48,6}*1728b, {24,12}*1728d
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
      6 facets:
         6 of {24}*48
      40 vertex figures:
         16 of {6}*12
         24 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      10 facets:
         6 of {8}*16
         4 of {24}*48
      24 vertex figures:
         24 of {6}*12

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