Polytope of Type {4,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4}*1800
Also Known As : {4,4}(15,0), {4,4|15}. if this polytope has another name.
Group : SmallGroup(1800,664)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 225, 450, 225
Order of s0s1s2 : 30
Order of s0s1s2s1 : 15
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   9-fold quotients : {4,4}*200
   25-fold quotients : {4,4}*72
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      75 facets:
         75 of {4}*8
      75 vertex figures:
         75 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 3.
      75 facets:
         75 of {4}*8
      75 vertex figures:
         75 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 5.
      45 facets:
         45 of {4}*8
      45 vertex figures:
         45 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 5.
      45 facets:
         45 of {4}*8
      45 vertex figures:
         45 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 5.
      45 facets:
         45 of {4}*8
      45 vertex figures:
         45 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 15.
      15 facets:
         15 of {4}*8
      15 vertex figures:
         15 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 15.
      15 facets:
         15 of {4}*8
      15 vertex figures:
         15 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 15.
      15 facets:
         15 of {4}*8
      15 vertex figures:
         15 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 15.
      15 facets:
         15 of {4}*8
      15 vertex figures:
         15 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 15.
      15 facets:
         15 of {4}*8
      15 vertex figures:
         15 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 15.
      15 facets:
         15 of {4}*8
      15 vertex figures:
         15 of {4}*8

Permutation Representation (GAP) : 
s0 := (  2, 12)(  3, 23)(  4,  9)(  5, 20)(  6, 21)(  8, 18)( 10, 15)( 11, 16)( 14, 24)( 17, 22)( 27, 37)( 28, 48)( 29, 34)( 30, 45)( 31, 46)( 33, 43)( 35, 40)( 36, 41)( 39, 49)( 42, 47)( 52, 62)( 53, 73)( 54, 59)( 55, 70)( 56, 71)( 58, 68)( 60, 65)( 61, 66)( 64, 74)( 67, 72)( 76,151)( 77,162)( 78,173)( 79,159)( 80,170)( 81,171)( 82,157)( 83,168)( 84,154)( 85,165)( 86,166)( 87,152)( 88,163)( 89,174)( 90,160)( 91,161)( 92,172)( 93,158)( 94,169)( 95,155)( 96,156)( 97,167)( 98,153)( 99,164)(100,175)(101,176)(102,187)(103,198)(104,184)(105,195)(106,196)(107,182)(108,193)(109,179)(110,190)(111,191)(112,177)(113,188)(114,199)(115,185)(116,186)(117,197)(118,183)(119,194)(120,180)(121,181)(122,192)(123,178)(124,189)(125,200)(126,201)(127,212)(128,223)(129,209)(130,220)(131,221)(132,207)(133,218)(134,204)(135,215)(136,216)(137,202)(138,213)(139,224)(140,210)(141,211)(142,222)(143,208)(144,219)(145,205)(146,206)(147,217)(148,203)(149,214)(150,225);;
s1 := (  2,  9)(  3, 12)(  4, 20)(  5, 23)(  6, 13)(  7, 16)(  8, 24)( 11, 25)( 15, 17)( 19, 21)( 26,151)( 27,159)( 28,162)( 29,170)( 30,173)( 31,163)( 32,166)( 33,174)( 34,152)( 35,160)( 36,175)( 37,153)( 38,156)( 39,164)( 40,167)( 41,157)( 42,165)( 43,168)( 44,171)( 45,154)( 46,169)( 47,172)( 48,155)( 49,158)( 50,161)( 51, 76)( 52, 84)( 53, 87)( 54, 95)( 55, 98)( 56, 88)( 57, 91)( 58, 99)( 59, 77)( 60, 85)( 61,100)( 62, 78)( 63, 81)( 64, 89)( 65, 92)( 66, 82)( 67, 90)( 68, 93)( 69, 96)( 70, 79)( 71, 94)( 72, 97)( 73, 80)( 74, 83)( 75, 86)(101,201)(102,209)(103,212)(104,220)(105,223)(106,213)(107,216)(108,224)(109,202)(110,210)(111,225)(112,203)(113,206)(114,214)(115,217)(116,207)(117,215)(118,218)(119,221)(120,204)(121,219)(122,222)(123,205)(124,208)(125,211)(127,134)(128,137)(129,145)(130,148)(131,138)(132,141)(133,149)(136,150)(140,142)(144,146)(177,184)(178,187)(179,195)(180,198)(181,188)(182,191)(183,199)(186,200)(190,192)(194,196);;
s2 := (  1, 32)(  2, 46)(  3, 40)(  4, 29)(  5, 43)(  6, 37)(  7, 26)(  8, 45)(  9, 34)( 10, 48)( 11, 42)( 12, 31)( 13, 50)( 14, 39)( 15, 28)( 16, 47)( 17, 36)( 18, 30)( 19, 44)( 20, 33)( 21, 27)( 22, 41)( 23, 35)( 24, 49)( 25, 38)( 51, 57)( 52, 71)( 53, 65)( 55, 68)( 56, 62)( 58, 70)( 60, 73)( 61, 67)( 63, 75)( 66, 72)( 76,107)( 77,121)( 78,115)( 79,104)( 80,118)( 81,112)( 82,101)( 83,120)( 84,109)( 85,123)( 86,117)( 87,106)( 88,125)( 89,114)( 90,103)( 91,122)( 92,111)( 93,105)( 94,119)( 95,108)( 96,102)( 97,116)( 98,110)( 99,124)(100,113)(126,132)(127,146)(128,140)(130,143)(131,137)(133,145)(135,148)(136,142)(138,150)(141,147)(151,182)(152,196)(153,190)(154,179)(155,193)(156,187)(157,176)(158,195)(159,184)(160,198)(161,192)(162,181)(163,200)(164,189)(165,178)(166,197)(167,186)(168,180)(169,194)(170,183)(171,177)(172,191)(173,185)(174,199)(175,188)(201,207)(202,221)(203,215)(205,218)(206,212)(208,220)(210,223)(211,217)(213,225)(216,222);;
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*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(225)!(  2, 12)(  3, 23)(  4,  9)(  5, 20)(  6, 21)(  8, 18)( 10, 15)( 11, 16)( 14, 24)( 17, 22)( 27, 37)( 28, 48)( 29, 34)( 30, 45)( 31, 46)( 33, 43)( 35, 40)( 36, 41)( 39, 49)( 42, 47)( 52, 62)( 53, 73)( 54, 59)( 55, 70)( 56, 71)( 58, 68)( 60, 65)( 61, 66)( 64, 74)( 67, 72)( 76,151)( 77,162)( 78,173)( 79,159)( 80,170)( 81,171)( 82,157)( 83,168)( 84,154)( 85,165)( 86,166)( 87,152)( 88,163)( 89,174)( 90,160)( 91,161)( 92,172)( 93,158)( 94,169)( 95,155)( 96,156)( 97,167)( 98,153)( 99,164)(100,175)(101,176)(102,187)(103,198)(104,184)(105,195)(106,196)(107,182)(108,193)(109,179)(110,190)(111,191)(112,177)(113,188)(114,199)(115,185)(116,186)(117,197)(118,183)(119,194)(120,180)(121,181)(122,192)(123,178)(124,189)(125,200)(126,201)(127,212)(128,223)(129,209)(130,220)(131,221)(132,207)(133,218)(134,204)(135,215)(136,216)(137,202)(138,213)(139,224)(140,210)(141,211)(142,222)(143,208)(144,219)(145,205)(146,206)(147,217)(148,203)(149,214)(150,225);
s1 := Sym(225)!(  2,  9)(  3, 12)(  4, 20)(  5, 23)(  6, 13)(  7, 16)(  8, 24)( 11, 25)( 15, 17)( 19, 21)( 26,151)( 27,159)( 28,162)( 29,170)( 30,173)( 31,163)( 32,166)( 33,174)( 34,152)( 35,160)( 36,175)( 37,153)( 38,156)( 39,164)( 40,167)( 41,157)( 42,165)( 43,168)( 44,171)( 45,154)( 46,169)( 47,172)( 48,155)( 49,158)( 50,161)( 51, 76)( 52, 84)( 53, 87)( 54, 95)( 55, 98)( 56, 88)( 57, 91)( 58, 99)( 59, 77)( 60, 85)( 61,100)( 62, 78)( 63, 81)( 64, 89)( 65, 92)( 66, 82)( 67, 90)( 68, 93)( 69, 96)( 70, 79)( 71, 94)( 72, 97)( 73, 80)( 74, 83)( 75, 86)(101,201)(102,209)(103,212)(104,220)(105,223)(106,213)(107,216)(108,224)(109,202)(110,210)(111,225)(112,203)(113,206)(114,214)(115,217)(116,207)(117,215)(118,218)(119,221)(120,204)(121,219)(122,222)(123,205)(124,208)(125,211)(127,134)(128,137)(129,145)(130,148)(131,138)(132,141)(133,149)(136,150)(140,142)(144,146)(177,184)(178,187)(179,195)(180,198)(181,188)(182,191)(183,199)(186,200)(190,192)(194,196);
s2 := Sym(225)!(  1, 32)(  2, 46)(  3, 40)(  4, 29)(  5, 43)(  6, 37)(  7, 26)(  8, 45)(  9, 34)( 10, 48)( 11, 42)( 12, 31)( 13, 50)( 14, 39)( 15, 28)( 16, 47)( 17, 36)( 18, 30)( 19, 44)( 20, 33)( 21, 27)( 22, 41)( 23, 35)( 24, 49)( 25, 38)( 51, 57)( 52, 71)( 53, 65)( 55, 68)( 56, 62)( 58, 70)( 60, 73)( 61, 67)( 63, 75)( 66, 72)( 76,107)( 77,121)( 78,115)( 79,104)( 80,118)( 81,112)( 82,101)( 83,120)( 84,109)( 85,123)( 86,117)( 87,106)( 88,125)( 89,114)( 90,103)( 91,122)( 92,111)( 93,105)( 94,119)( 95,108)( 96,102)( 97,116)( 98,110)( 99,124)(100,113)(126,132)(127,146)(128,140)(130,143)(131,137)(133,145)(135,148)(136,142)(138,150)(141,147)(151,182)(152,196)(153,190)(154,179)(155,193)(156,187)(157,176)(158,195)(159,184)(160,198)(161,192)(162,181)(163,200)(164,189)(165,178)(166,197)(167,186)(168,180)(169,194)(170,183)(171,177)(172,191)(173,185)(174,199)(175,188)(201,207)(202,221)(203,215)(205,218)(206,212)(208,220)(210,223)(211,217)(213,225)(216,222);
poly := sub<Sym(225)|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*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope

Twisty Puzzle