Polytope of Type {4,78}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,78}*1872
if this polytope has a name.
Group : SmallGroup(1872,1023)
Rank : 3
Schlafli Type : {4,78}
Number of vertices, edges, etc : 12, 468, 234
Order of s₀s₁s₂ : 52
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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) :
   9-fold quotients : {4,26}*208
   13-fold quotients : {4,6}*144
   18-fold quotients : {2,26}*104
   26-fold quotients : {4,6}*72
   36-fold quotients : {2,13}*52
   117-fold quotients : {4,2}*16
   234-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope