Polytope of Type {6,78}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,78}*1404
if this polytope has a name.
Group : SmallGroup(1404,122)
Rank : 3
Schlafli Type : {6,78}
Number of vertices, edges, etc : 9, 351, 117
Order of s₀s₁s₂ : 39
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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) :
   13-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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