Polytope of Type {6,78}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,78}*936a
if this polytope has a name.
Group : SmallGroup(936,189)
Rank : 3
Schlafli Type : {6,78}
Number of vertices, edges, etc : 6, 234, 78
Order of s₀s₁s₂ : 78
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,78,2} of size 1872
Vertex Figure Of :
   {2,6,78} of size 1872
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,26}*312
   9-fold quotients : {2,26}*104
   13-fold quotients : {6,6}*72c
   18-fold quotients : {2,13}*52
   26-fold quotients : {3,6}*36
   39-fold quotients : {6,2}*24
   78-fold quotients : {3,2}*12
   117-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,156}*1872a, {12,78}*1872a
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope