Polytope of Type {78,6}

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

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

Permutation Representation (Magma) :

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

References : None.
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