Polytope of Type {4,40}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,40}*1600b
if this polytope has a name.
Group : SmallGroup(1600,6690)
Rank : 3
Schlafli Type : {4,40}
Number of vertices, edges, etc : 20, 400, 200
Order of s0s1s2 : 8
Order of s0s1s2s1 : 20
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 : {4,20}*800
   4-fold quotients : {4,10}*400
   8-fold quotients : {4,10}*200
   25-fold quotients : {4,8}*64b
   50-fold quotients : {4,4}*32
   100-fold quotients : {2,4}*16, {4,2}*16
   200-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*s2*s1*s2*s1*s2*s1*s2*s1> of order 5.
      40 facets:
         40 of {4}*8
      4 vertex figures:
         4 of {40}*80
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 5.
      40 facets:
         40 of {4}*8
      4 vertex figures:
         4 of {40}*80
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 5.
      40 facets:
         40 of {4}*8
      12 vertex figures:
         10 of {8}*16
         2 of {40}*80

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