Polytope of Type {2,78,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,78,6}*1872a
if this polytope has a name.
Group : SmallGroup(1872,1061)
Rank : 4
Schlafli Type : {2,78,6}
Number of vertices, edges, etc : 2, 78, 234, 6
Order of s₀s₁s₂s₃ : 78
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,26,6}*624
   9-fold quotients : {2,26,2}*208
   13-fold quotients : {2,6,6}*144b
   18-fold quotients : {2,13,2}*104
   26-fold quotients : {2,6,3}*72
   39-fold quotients : {2,2,6}*48
   78-fold quotients : {2,2,3}*24
   117-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)(  9, 10)( 16, 29)( 17, 41)
( 18, 40)( 19, 39)( 20, 38)( 21, 37)( 22, 36)( 23, 35)( 24, 34)( 25, 33)
( 26, 32)( 27, 31)( 28, 30)( 43, 54)( 44, 53)( 45, 52)( 46, 51)( 47, 50)
( 48, 49)( 55, 68)( 56, 80)( 57, 79)( 58, 78)( 59, 77)( 60, 76)( 61, 75)
( 62, 74)( 63, 73)( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 82, 93)( 83, 92)
( 84, 91)( 85, 90)( 86, 89)( 87, 88)( 94,107)( 95,119)( 96,118)( 97,117)
( 98,116)( 99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)
(106,108);;
s2 := (  3, 17)(  4, 16)(  5, 28)(  6, 27)(  7, 26)(  8, 25)(  9, 24)( 10, 23)
( 11, 22)( 12, 21)( 13, 20)( 14, 19)( 15, 18)( 29, 30)( 31, 41)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 42, 95)( 43, 94)( 44,106)( 45,105)( 46,104)
( 47,103)( 48,102)( 49,101)( 50,100)( 51, 99)( 52, 98)( 53, 97)( 54, 96)
( 55, 82)( 56, 81)( 57, 93)( 58, 92)( 59, 91)( 60, 90)( 61, 89)( 62, 88)
( 63, 87)( 64, 86)( 65, 85)( 66, 84)( 67, 83)( 68,108)( 69,107)( 70,119)
( 71,118)( 72,117)( 73,116)( 74,115)( 75,114)( 76,113)( 77,112)( 78,111)
( 79,110)( 80,109);;
s3 := (  3, 42)(  4, 43)(  5, 44)(  6, 45)(  7, 46)(  8, 47)(  9, 48)( 10, 49)
( 11, 50)( 12, 51)( 13, 52)( 14, 53)( 15, 54)( 16, 68)( 17, 69)( 18, 70)
( 19, 71)( 20, 72)( 21, 73)( 22, 74)( 23, 75)( 24, 76)( 25, 77)( 26, 78)
( 27, 79)( 28, 80)( 29, 55)( 30, 56)( 31, 57)( 32, 58)( 33, 59)( 34, 60)
( 35, 61)( 36, 62)( 37, 63)( 38, 64)( 39, 65)( 40, 66)( 41, 67)( 94,107)
( 95,108)( 96,109)( 97,110)( 98,111)( 99,112)(100,113)(101,114)(102,115)
(103,116)(104,117)(105,118)(106,119);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s3*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(119)!(1,2);
s1 := Sym(119)!(  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)(  9, 10)( 16, 29)
( 17, 41)( 18, 40)( 19, 39)( 20, 38)( 21, 37)( 22, 36)( 23, 35)( 24, 34)
( 25, 33)( 26, 32)( 27, 31)( 28, 30)( 43, 54)( 44, 53)( 45, 52)( 46, 51)
( 47, 50)( 48, 49)( 55, 68)( 56, 80)( 57, 79)( 58, 78)( 59, 77)( 60, 76)
( 61, 75)( 62, 74)( 63, 73)( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 82, 93)
( 83, 92)( 84, 91)( 85, 90)( 86, 89)( 87, 88)( 94,107)( 95,119)( 96,118)
( 97,117)( 98,116)( 99,115)(100,114)(101,113)(102,112)(103,111)(104,110)
(105,109)(106,108);
s2 := Sym(119)!(  3, 17)(  4, 16)(  5, 28)(  6, 27)(  7, 26)(  8, 25)(  9, 24)
( 10, 23)( 11, 22)( 12, 21)( 13, 20)( 14, 19)( 15, 18)( 29, 30)( 31, 41)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 42, 95)( 43, 94)( 44,106)( 45,105)
( 46,104)( 47,103)( 48,102)( 49,101)( 50,100)( 51, 99)( 52, 98)( 53, 97)
( 54, 96)( 55, 82)( 56, 81)( 57, 93)( 58, 92)( 59, 91)( 60, 90)( 61, 89)
( 62, 88)( 63, 87)( 64, 86)( 65, 85)( 66, 84)( 67, 83)( 68,108)( 69,107)
( 70,119)( 71,118)( 72,117)( 73,116)( 74,115)( 75,114)( 76,113)( 77,112)
( 78,111)( 79,110)( 80,109);
s3 := Sym(119)!(  3, 42)(  4, 43)(  5, 44)(  6, 45)(  7, 46)(  8, 47)(  9, 48)
( 10, 49)( 11, 50)( 12, 51)( 13, 52)( 14, 53)( 15, 54)( 16, 68)( 17, 69)
( 18, 70)( 19, 71)( 20, 72)( 21, 73)( 22, 74)( 23, 75)( 24, 76)( 25, 77)
( 26, 78)( 27, 79)( 28, 80)( 29, 55)( 30, 56)( 31, 57)( 32, 58)( 33, 59)
( 34, 60)( 35, 61)( 36, 62)( 37, 63)( 38, 64)( 39, 65)( 40, 66)( 41, 67)
( 94,107)( 95,108)( 96,109)( 97,110)( 98,111)( 99,112)(100,113)(101,114)
(102,115)(103,116)(104,117)(105,118)(106,119);
poly := sub<Sym(119)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s3*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 >;

to this polytope