Polytope of Type {55,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {55,4}*1760
if this polytope has a name.
Group : SmallGroup(1760,1271)
Rank : 3
Schlafli Type : {55,4}
Number of vertices, edges, etc : 220, 440, 16
Order of s0s1s2 : 55
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Self-Petrie
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) :
   11-fold quotients : {5,4}*160
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1> of order 2.
      8 facets:
         8 of {55}*110
      110 vertex figures:
         110 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 2.
      8 facets:
         8 of {55}*110
      132 vertex figures:
         88 of {4}*8
         44 of {2}*4
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      4 facets:
         4 of {55}*110
      77 vertex figures:
         33 of {4}*8
         44 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
      4 facets:
         4 of {55}*110
      66 vertex figures:
         44 of {4}*8
         22 of {2}*4

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