Polytope of Type {24,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,4}*1728a
if this polytope has a name.
Group : SmallGroup(1728,12703)
Rank : 3
Schlafli Type : {24,4}
Number of vertices, edges, etc : 216, 432, 36
Order of s0s1s2 : 24
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,4}*864b
   3-fold quotients : {24,4}*576a
   4-fold quotients : {6,4}*432a
   6-fold quotients : {12,4}*288
   8-fold quotients : {6,4}*216
   12-fold quotients : {6,4}*144
   24-fold quotients : {6,4}*72
   27-fold quotients : {8,4}*64a
   54-fold quotients : {4,4}*32, {8,2}*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=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      18 facets:
         18 of {24}*48
      108 vertex figures:
         108 of {4}*8
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      18 facets:
         18 of {24}*48
      120 vertex figures:
         96 of {4}*8
         24 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
      16 facets:
         10 of {24}*48
         6 of {8}*16
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 3.
      12 facets:
         12 of {24}*48
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2> of order 6.
      8 facets:
         3 of {8}*16
         5 of {24}*48
      48 vertex figures:
         24 of {4}*8
         24 of {2}*4
   P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2> of order 6.
      6 facets:
         6 of {24}*48
      48 vertex figures:
         24 of {2}*4
         24 of {4}*8

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