Polytope of Type {78,6}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope