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}*1600a
if this polytope has a name.
Group : SmallGroup(1600,6672)
Rank : 3
Schlafli Type : {40,4}
Number of vertices, edges, etc : 200, 400, 20
Order of s0s1s2 : 8
Order of s0s1s2s1 : 10
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 : {20,4}*800
   4-fold quotients : {10,4}*400
   8-fold quotients : {10,4}*200
   25-fold quotients : {8,4}*64a
   50-fold quotients : {4,4}*32, {8,2}*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*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 2.
      10 facets:
         10 of {40}*80
      100 vertex figures:
         100 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 5.
      4 facets:
         4 of {40}*80
      40 vertex figures:
         40 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*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 := (  1,101)(  2,105)(  3,104)(  4,103)(  5,102)(  6,121)(  7,125)(  8,124)(  9,123)( 10,122)( 11,116)( 12,120)( 13,119)( 14,118)( 15,117)( 16,111)( 17,115)( 18,114)( 19,113)( 20,112)( 21,106)( 22,110)( 23,109)( 24,108)( 25,107)( 26,126)( 27,130)( 28,129)( 29,128)( 30,127)( 31,146)( 32,150)( 33,149)( 34,148)( 35,147)( 36,141)( 37,145)( 38,144)( 39,143)( 40,142)( 41,136)( 42,140)( 43,139)( 44,138)( 45,137)( 46,131)( 47,135)( 48,134)( 49,133)( 50,132)( 51,176)( 52,180)( 53,179)( 54,178)( 55,177)( 56,196)( 57,200)( 58,199)( 59,198)( 60,197)( 61,191)( 62,195)( 63,194)( 64,193)( 65,192)( 66,186)( 67,190)( 68,189)( 69,188)( 70,187)( 71,181)( 72,185)( 73,184)( 74,183)( 75,182)( 76,151)( 77,155)( 78,154)( 79,153)( 80,152)( 81,171)( 82,175)( 83,174)( 84,173)( 85,172)( 86,166)( 87,170)( 88,169)( 89,168)( 90,167)( 91,161)( 92,165)( 93,164)( 94,163)( 95,162)( 96,156)( 97,160)( 98,159)( 99,158)(100,157);;
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)( 52, 62)( 53, 73)( 54, 59)( 55, 70)( 56, 71)( 58, 68)( 60, 65)( 61, 66)( 64, 74)( 67, 72)( 77, 87)( 78, 98)( 79, 84)( 80, 95)( 81, 96)( 83, 93)( 85, 90)( 86, 91)( 89, 99)( 92, 97)(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)(152,162)(153,173)(154,159)(155,170)(156,171)(158,168)(160,165)(161,166)(164,174)(167,172)(177,187)(178,198)(179,184)(180,195)(181,196)(183,193)(185,190)(186,191)(189,199)(192,197);;
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*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(200)!(  1,101)(  2,105)(  3,104)(  4,103)(  5,102)(  6,121)(  7,125)(  8,124)(  9,123)( 10,122)( 11,116)( 12,120)( 13,119)( 14,118)( 15,117)( 16,111)( 17,115)( 18,114)( 19,113)( 20,112)( 21,106)( 22,110)( 23,109)( 24,108)( 25,107)( 26,126)( 27,130)( 28,129)( 29,128)( 30,127)( 31,146)( 32,150)( 33,149)( 34,148)( 35,147)( 36,141)( 37,145)( 38,144)( 39,143)( 40,142)( 41,136)( 42,140)( 43,139)( 44,138)( 45,137)( 46,131)( 47,135)( 48,134)( 49,133)( 50,132)( 51,176)( 52,180)( 53,179)( 54,178)( 55,177)( 56,196)( 57,200)( 58,199)( 59,198)( 60,197)( 61,191)( 62,195)( 63,194)( 64,193)( 65,192)( 66,186)( 67,190)( 68,189)( 69,188)( 70,187)( 71,181)( 72,185)( 73,184)( 74,183)( 75,182)( 76,151)( 77,155)( 78,154)( 79,153)( 80,152)( 81,171)( 82,175)( 83,174)( 84,173)( 85,172)( 86,166)( 87,170)( 88,169)( 89,168)( 90,167)( 91,161)( 92,165)( 93,164)( 94,163)( 95,162)( 96,156)( 97,160)( 98,159)( 99,158)(100,157);
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)( 52, 62)( 53, 73)( 54, 59)( 55, 70)( 56, 71)( 58, 68)( 60, 65)( 61, 66)( 64, 74)( 67, 72)( 77, 87)( 78, 98)( 79, 84)( 80, 95)( 81, 96)( 83, 93)( 85, 90)( 86, 91)( 89, 99)( 92, 97)(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)(152,162)(153,173)(154,159)(155,170)(156,171)(158,168)(160,165)(161,166)(164,174)(167,172)(177,187)(178,198)(179,184)(180,195)(181,196)(183,193)(185,190)(186,191)(189,199)(192,197);
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*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1 >;
References : None.
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