Polytope of Type {40,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {40,4}*1600b
if this polytope has a name.
Group : SmallGroup(1600,6690)
Rank : 3
Schlafli Type : {40,4}
Number of vertices, edges, etc : 200, 400, 20
Order of s0s1s2 : 8
Order of s0s1s2s1 : 20
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {20,4}*800
   4-fold quotients : {10,4}*400
   8-fold quotients : {10,4}*200
   25-fold quotients : {8,4}*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*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 5.
      4 facets:
         4 of {40}*80
      40 vertex figures:
         40 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 5.
      4 facets:
         4 of {40}*80
      40 vertex figures:
         40 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 5.
      12 facets:
         10 of {8}*16
         2 of {40}*80
      40 vertex figures:
         40 of {4}*8

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