Polytope of Type {58,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {58,4}*464
Also Known As : {58,4|2}. if this polytope has another name.
Group : SmallGroup(464,39)
Rank : 3
Schlafli Type : {58,4}
Number of vertices, edges, etc : 58, 116, 4
Order of s₀s₁s₂ : 116
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {58,4,2} of size 928
   {58,4,4} of size 1856
Vertex Figure Of :
   {2,58,4} of size 928
   {4,58,4} of size 1856
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {58,2}*232
   4-fold quotients : {29,2}*116
   29-fold quotients : {2,4}*16
   58-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {116,4}*928, {58,8}*928
   3-fold covers : {58,12}*1392, {174,4}*1392a
   4-fold covers : {116,8}*1856a, {232,4}*1856a, {116,8}*1856b, {232,4}*1856b, {116,4}*1856, {58,16}*1856
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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