Polytope of Type {39,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {39,6}*1404
if this polytope has a name.
Group : SmallGroup(1404,122)
Rank : 3
Schlafli Type : {39,6}
Number of vertices, edges, etc : 117, 351, 18
Order of s0s1s2 : 78
Order of s0s1s2s1 : 6
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) :
   3-fold quotients : {39,6}*468
   9-fold quotients : {39,2}*156
   13-fold quotients : {3,6}*108
   27-fold quotients : {13,2}*52
   39-fold quotients : {3,6}*36
   117-fold quotients : {3,2}*12
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> of order 3.
      6 facets:
         6 of {39}*78
      65 vertex figures:
         26 of {6}*12
         39 of {2}*4

Permutation Representation (GAP) : 
s0 := (  4, 37)(  5, 38)(  6, 39)(  7, 34)(  8, 35)(  9, 36)( 10, 31)( 11, 32)( 12, 33)( 13, 28)( 14, 29)( 15, 30)( 16, 25)( 17, 26)( 18, 27)( 19, 22)( 20, 23)( 21, 24)( 40, 79)( 41, 80)( 42, 81)( 43,115)( 44,116)( 45,117)( 46,112)( 47,113)( 48,114)( 49,109)( 50,110)( 51,111)( 52,106)( 53,107)( 54,108)( 55,103)( 56,104)( 57,105)( 58,100)( 59,101)( 60,102)( 61, 97)( 62, 98)( 63, 99)( 64, 94)( 65, 95)( 66, 96)( 67, 91)( 68, 92)( 69, 93)( 70, 88)( 71, 89)( 72, 90)( 73, 85)( 74, 86)( 75, 87)( 76, 82)( 77, 83)( 78, 84);;
s1 := (  1, 44)(  2, 45)(  3, 43)(  4, 41)(  5, 42)(  6, 40)(  7, 77)(  8, 78)(  9, 76)( 10, 74)( 11, 75)( 12, 73)( 13, 71)( 14, 72)( 15, 70)( 16, 68)( 17, 69)( 18, 67)( 19, 65)( 20, 66)( 21, 64)( 22, 62)( 23, 63)( 24, 61)( 25, 59)( 26, 60)( 27, 58)( 28, 56)( 29, 57)( 30, 55)( 31, 53)( 32, 54)( 33, 52)( 34, 50)( 35, 51)( 36, 49)( 37, 47)( 38, 48)( 39, 46)( 79, 82)( 80, 83)( 81, 84)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)( 91,109)( 92,110)( 93,111)( 94,106)( 95,107)( 96,108)( 97,103)( 98,104)( 99,105);;
s2 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 40, 79)( 41, 81)( 42, 80)( 43, 82)( 44, 84)( 45, 83)( 46, 85)( 47, 87)( 48, 86)( 49, 88)( 50, 90)( 51, 89)( 52, 91)( 53, 93)( 54, 92)( 55, 94)( 56, 96)( 57, 95)( 58, 97)( 59, 99)( 60, 98)( 61,100)( 62,102)( 63,101)( 64,103)( 65,105)( 66,104)( 67,106)( 68,108)( 69,107)( 70,109)( 71,111)( 72,110)( 73,112)( 74,114)( 75,113)( 76,115)( 77,117)( 78,116);;
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*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s0, 
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(117)!(  4, 37)(  5, 38)(  6, 39)(  7, 34)(  8, 35)(  9, 36)( 10, 31)( 11, 32)( 12, 33)( 13, 28)( 14, 29)( 15, 30)( 16, 25)( 17, 26)( 18, 27)( 19, 22)( 20, 23)( 21, 24)( 40, 79)( 41, 80)( 42, 81)( 43,115)( 44,116)( 45,117)( 46,112)( 47,113)( 48,114)( 49,109)( 50,110)( 51,111)( 52,106)( 53,107)( 54,108)( 55,103)( 56,104)( 57,105)( 58,100)( 59,101)( 60,102)( 61, 97)( 62, 98)( 63, 99)( 64, 94)( 65, 95)( 66, 96)( 67, 91)( 68, 92)( 69, 93)( 70, 88)( 71, 89)( 72, 90)( 73, 85)( 74, 86)( 75, 87)( 76, 82)( 77, 83)( 78, 84);
s1 := Sym(117)!(  1, 44)(  2, 45)(  3, 43)(  4, 41)(  5, 42)(  6, 40)(  7, 77)(  8, 78)(  9, 76)( 10, 74)( 11, 75)( 12, 73)( 13, 71)( 14, 72)( 15, 70)( 16, 68)( 17, 69)( 18, 67)( 19, 65)( 20, 66)( 21, 64)( 22, 62)( 23, 63)( 24, 61)( 25, 59)( 26, 60)( 27, 58)( 28, 56)( 29, 57)( 30, 55)( 31, 53)( 32, 54)( 33, 52)( 34, 50)( 35, 51)( 36, 49)( 37, 47)( 38, 48)( 39, 46)( 79, 82)( 80, 83)( 81, 84)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)( 91,109)( 92,110)( 93,111)( 94,106)( 95,107)( 96,108)( 97,103)( 98,104)( 99,105);
s2 := Sym(117)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 40, 79)( 41, 81)( 42, 80)( 43, 82)( 44, 84)( 45, 83)( 46, 85)( 47, 87)( 48, 86)( 49, 88)( 50, 90)( 51, 89)( 52, 91)( 53, 93)( 54, 92)( 55, 94)( 56, 96)( 57, 95)( 58, 97)( 59, 99)( 60, 98)( 61,100)( 62,102)( 63,101)( 64,103)( 65,105)( 66,104)( 67,106)( 68,108)( 69,107)( 70,109)( 71,111)( 72,110)( 73,112)( 74,114)( 75,113)( 76,115)( 77,117)( 78,116);
poly := sub<Sym(117)|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*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s0, 
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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