Polytope of Type {138,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {138,6}*1656a
if this polytope has a name.
Group : SmallGroup(1656,116)
Rank : 3
Schlafli Type : {138,6}
Number of vertices, edges, etc : 138, 414, 6
Order of s0s1s2 : 138
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
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) :
   3-fold quotients : {46,6}*552
   9-fold quotients : {46,2}*184
   18-fold quotients : {23,2}*92
   23-fold quotients : {6,6}*72b
   46-fold quotients : {6,3}*36
   69-fold quotients : {2,6}*24
   138-fold quotients : {2,3}*12
   207-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   None.

Permutation Representation (GAP) : 
s0 := (  2, 23)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 18)(  8, 17)(  9, 16)( 10, 15)( 11, 14)( 12, 13)( 25, 46)( 26, 45)( 27, 44)( 28, 43)( 29, 42)( 30, 41)( 31, 40)( 32, 39)( 33, 38)( 34, 37)( 35, 36)( 48, 69)( 49, 68)( 50, 67)( 51, 66)( 52, 65)( 53, 64)( 54, 63)( 55, 62)( 56, 61)( 57, 60)( 58, 59)( 70,139)( 71,161)( 72,160)( 73,159)( 74,158)( 75,157)( 76,156)( 77,155)( 78,154)( 79,153)( 80,152)( 81,151)( 82,150)( 83,149)( 84,148)( 85,147)( 86,146)( 87,145)( 88,144)( 89,143)( 90,142)( 91,141)( 92,140)( 93,162)( 94,184)( 95,183)( 96,182)( 97,181)( 98,180)( 99,179)(100,178)(101,177)(102,176)(103,175)(104,174)(105,173)(106,172)(107,171)(108,170)(109,169)(110,168)(111,167)(112,166)(113,165)(114,164)(115,163)(116,185)(117,207)(118,206)(119,205)(120,204)(121,203)(122,202)(123,201)(124,200)(125,199)(126,198)(127,197)(128,196)(129,195)(130,194)(131,193)(132,192)(133,191)(134,190)(135,189)(136,188)(137,187)(138,186);;
s1 := (  1, 71)(  2, 70)(  3, 92)(  4, 91)(  5, 90)(  6, 89)(  7, 88)(  8, 87)(  9, 86)( 10, 85)( 11, 84)( 12, 83)( 13, 82)( 14, 81)( 15, 80)( 16, 79)( 17, 78)( 18, 77)( 19, 76)( 20, 75)( 21, 74)( 22, 73)( 23, 72)( 24,117)( 25,116)( 26,138)( 27,137)( 28,136)( 29,135)( 30,134)( 31,133)( 32,132)( 33,131)( 34,130)( 35,129)( 36,128)( 37,127)( 38,126)( 39,125)( 40,124)( 41,123)( 42,122)( 43,121)( 44,120)( 45,119)( 46,118)( 47, 94)( 48, 93)( 49,115)( 50,114)( 51,113)( 52,112)( 53,111)( 54,110)( 55,109)( 56,108)( 57,107)( 58,106)( 59,105)( 60,104)( 61,103)( 62,102)( 63,101)( 64,100)( 65, 99)( 66, 98)( 67, 97)( 68, 96)( 69, 95)(139,140)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152)(162,186)(163,185)(164,207)(165,206)(166,205)(167,204)(168,203)(169,202)(170,201)(171,200)(172,199)(173,198)(174,197)(175,196)(176,195)(177,194)(178,193)(179,192)(180,191)(181,190)(182,189)(183,188)(184,187);;
s2 := (  1, 24)(  2, 25)(  3, 26)(  4, 27)(  5, 28)(  6, 29)(  7, 30)(  8, 31)(  9, 32)( 10, 33)( 11, 34)( 12, 35)( 13, 36)( 14, 37)( 15, 38)( 16, 39)( 17, 40)( 18, 41)( 19, 42)( 20, 43)( 21, 44)( 22, 45)( 23, 46)( 70,162)( 71,163)( 72,164)( 73,165)( 74,166)( 75,167)( 76,168)( 77,169)( 78,170)( 79,171)( 80,172)( 81,173)( 82,174)( 83,175)( 84,176)( 85,177)( 86,178)( 87,179)( 88,180)( 89,181)( 90,182)( 91,183)( 92,184)( 93,139)( 94,140)( 95,141)( 96,142)( 97,143)( 98,144)( 99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,161)(116,185)(117,186)(118,187)(119,188)(120,189)(121,190)(122,191)(123,192)(124,193)(125,194)(126,195)(127,196)(128,197)(129,198)(130,199)(131,200)(132,201)(133,202)(134,203)(135,204)(136,205)(137,206)(138,207);;
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*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*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*s2*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*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(207)!(  2, 23)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 18)(  8, 17)(  9, 16)( 10, 15)( 11, 14)( 12, 13)( 25, 46)( 26, 45)( 27, 44)( 28, 43)( 29, 42)( 30, 41)( 31, 40)( 32, 39)( 33, 38)( 34, 37)( 35, 36)( 48, 69)( 49, 68)( 50, 67)( 51, 66)( 52, 65)( 53, 64)( 54, 63)( 55, 62)( 56, 61)( 57, 60)( 58, 59)( 70,139)( 71,161)( 72,160)( 73,159)( 74,158)( 75,157)( 76,156)( 77,155)( 78,154)( 79,153)( 80,152)( 81,151)( 82,150)( 83,149)( 84,148)( 85,147)( 86,146)( 87,145)( 88,144)( 89,143)( 90,142)( 91,141)( 92,140)( 93,162)( 94,184)( 95,183)( 96,182)( 97,181)( 98,180)( 99,179)(100,178)(101,177)(102,176)(103,175)(104,174)(105,173)(106,172)(107,171)(108,170)(109,169)(110,168)(111,167)(112,166)(113,165)(114,164)(115,163)(116,185)(117,207)(118,206)(119,205)(120,204)(121,203)(122,202)(123,201)(124,200)(125,199)(126,198)(127,197)(128,196)(129,195)(130,194)(131,193)(132,192)(133,191)(134,190)(135,189)(136,188)(137,187)(138,186);
s1 := Sym(207)!(  1, 71)(  2, 70)(  3, 92)(  4, 91)(  5, 90)(  6, 89)(  7, 88)(  8, 87)(  9, 86)( 10, 85)( 11, 84)( 12, 83)( 13, 82)( 14, 81)( 15, 80)( 16, 79)( 17, 78)( 18, 77)( 19, 76)( 20, 75)( 21, 74)( 22, 73)( 23, 72)( 24,117)( 25,116)( 26,138)( 27,137)( 28,136)( 29,135)( 30,134)( 31,133)( 32,132)( 33,131)( 34,130)( 35,129)( 36,128)( 37,127)( 38,126)( 39,125)( 40,124)( 41,123)( 42,122)( 43,121)( 44,120)( 45,119)( 46,118)( 47, 94)( 48, 93)( 49,115)( 50,114)( 51,113)( 52,112)( 53,111)( 54,110)( 55,109)( 56,108)( 57,107)( 58,106)( 59,105)( 60,104)( 61,103)( 62,102)( 63,101)( 64,100)( 65, 99)( 66, 98)( 67, 97)( 68, 96)( 69, 95)(139,140)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152)(162,186)(163,185)(164,207)(165,206)(166,205)(167,204)(168,203)(169,202)(170,201)(171,200)(172,199)(173,198)(174,197)(175,196)(176,195)(177,194)(178,193)(179,192)(180,191)(181,190)(182,189)(183,188)(184,187);
s2 := Sym(207)!(  1, 24)(  2, 25)(  3, 26)(  4, 27)(  5, 28)(  6, 29)(  7, 30)(  8, 31)(  9, 32)( 10, 33)( 11, 34)( 12, 35)( 13, 36)( 14, 37)( 15, 38)( 16, 39)( 17, 40)( 18, 41)( 19, 42)( 20, 43)( 21, 44)( 22, 45)( 23, 46)( 70,162)( 71,163)( 72,164)( 73,165)( 74,166)( 75,167)( 76,168)( 77,169)( 78,170)( 79,171)( 80,172)( 81,173)( 82,174)( 83,175)( 84,176)( 85,177)( 86,178)( 87,179)( 88,180)( 89,181)( 90,182)( 91,183)( 92,184)( 93,139)( 94,140)( 95,141)( 96,142)( 97,143)( 98,144)( 99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,161)(116,185)(117,186)(118,187)(119,188)(120,189)(121,190)(122,191)(123,192)(124,193)(125,194)(126,195)(127,196)(128,197)(129,198)(130,199)(131,200)(132,201)(133,202)(134,203)(135,204)(136,205)(137,206)(138,207);
poly := sub<Sym(207)|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*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*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*s2*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*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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