Polytope of Type {6,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,30}*480
if this polytope has a name.
Group : SmallGroup(480,1193)
Rank : 3
Schlafli Type : {6,30}
Number of vertices, edges, etc : 8, 120, 40
Order of s₀s₁s₂ : 20
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,30,2} of size 960
   {6,30,4} of size 1920
Vertex Figure Of :
   {2,6,30} of size 960
   {4,6,30} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,15}*240
   5-fold quotients : {6,6}*96
   10-fold quotients : {3,6}*48, {6,3}*48
   12-fold quotients : {2,10}*40
   20-fold quotients : {3,3}*24
   24-fold quotients : {2,5}*20
   60-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,60}*960a, {12,30}*960a, {6,30}*960, {6,60}*960b, {12,30}*960b
   3-fold covers : {6,30}*1440g, {6,30}*1440h
   4-fold covers : {6,30}*1920a, {12,60}*1920a, {12,60}*1920b, {6,60}*1920, {6,30}*1920b, {6,30}*1920c, {6,120}*1920a, {6,120}*1920b, {12,60}*1920c, {24,30}*1920a, {12,30}*1920, {12,60}*1920d, {24,30}*1920b
Permutation Representation (GAP) :

s0 := (  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)( 23, 42)
( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)( 31, 50)
( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)( 39, 58)
( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)( 82,103)
( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)( 90,111)
( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)( 98,119)
( 99,118)(100,120);;
s1 := (  1, 21)(  2, 22)(  3, 24)(  4, 23)(  5, 37)(  6, 38)(  7, 40)(  8, 39)
(  9, 33)( 10, 34)( 11, 36)( 12, 35)( 13, 29)( 14, 30)( 15, 32)( 16, 31)
( 17, 25)( 18, 26)( 19, 28)( 20, 27)( 43, 44)( 45, 57)( 46, 58)( 47, 60)
( 48, 59)( 49, 53)( 50, 54)( 51, 56)( 52, 55)( 61, 81)( 62, 82)( 63, 84)
( 64, 83)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69, 93)( 70, 94)( 71, 96)
( 72, 95)( 73, 89)( 74, 90)( 75, 92)( 76, 91)( 77, 85)( 78, 86)( 79, 88)
( 80, 87)(103,104)(105,117)(106,118)(107,120)(108,119)(109,113)(110,114)
(111,116)(112,115);;
s2 := (  1, 68)(  2, 66)(  3, 67)(  4, 65)(  5, 64)(  6, 62)(  7, 63)(  8, 61)
(  9, 80)( 10, 78)( 11, 79)( 12, 77)( 13, 76)( 14, 74)( 15, 75)( 16, 73)
( 17, 72)( 18, 70)( 19, 71)( 20, 69)( 21,108)( 22,106)( 23,107)( 24,105)
( 25,104)( 26,102)( 27,103)( 28,101)( 29,120)( 30,118)( 31,119)( 32,117)
( 33,116)( 34,114)( 35,115)( 36,113)( 37,112)( 38,110)( 39,111)( 40,109)
( 41, 88)( 42, 86)( 43, 87)( 44, 85)( 45, 84)( 46, 82)( 47, 83)( 48, 81)
( 49,100)( 50, 98)( 51, 99)( 52, 97)( 53, 96)( 54, 94)( 55, 95)( 56, 93)
( 57, 92)( 58, 90)( 59, 91)( 60, 89);;
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, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(120)!(  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)
( 23, 42)( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)
( 31, 50)( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)
( 39, 58)( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)
( 82,103)( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)
( 90,111)( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)
( 98,119)( 99,118)(100,120);
s1 := Sym(120)!(  1, 21)(  2, 22)(  3, 24)(  4, 23)(  5, 37)(  6, 38)(  7, 40)
(  8, 39)(  9, 33)( 10, 34)( 11, 36)( 12, 35)( 13, 29)( 14, 30)( 15, 32)
( 16, 31)( 17, 25)( 18, 26)( 19, 28)( 20, 27)( 43, 44)( 45, 57)( 46, 58)
( 47, 60)( 48, 59)( 49, 53)( 50, 54)( 51, 56)( 52, 55)( 61, 81)( 62, 82)
( 63, 84)( 64, 83)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69, 93)( 70, 94)
( 71, 96)( 72, 95)( 73, 89)( 74, 90)( 75, 92)( 76, 91)( 77, 85)( 78, 86)
( 79, 88)( 80, 87)(103,104)(105,117)(106,118)(107,120)(108,119)(109,113)
(110,114)(111,116)(112,115);
s2 := Sym(120)!(  1, 68)(  2, 66)(  3, 67)(  4, 65)(  5, 64)(  6, 62)(  7, 63)
(  8, 61)(  9, 80)( 10, 78)( 11, 79)( 12, 77)( 13, 76)( 14, 74)( 15, 75)
( 16, 73)( 17, 72)( 18, 70)( 19, 71)( 20, 69)( 21,108)( 22,106)( 23,107)
( 24,105)( 25,104)( 26,102)( 27,103)( 28,101)( 29,120)( 30,118)( 31,119)
( 32,117)( 33,116)( 34,114)( 35,115)( 36,113)( 37,112)( 38,110)( 39,111)
( 40,109)( 41, 88)( 42, 86)( 43, 87)( 44, 85)( 45, 84)( 46, 82)( 47, 83)
( 48, 81)( 49,100)( 50, 98)( 51, 99)( 52, 97)( 53, 96)( 54, 94)( 55, 95)
( 56, 93)( 57, 92)( 58, 90)( 59, 91)( 60, 89);
poly := sub<Sym(120)|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, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 >;

References : None.
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