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Polytope of Type {20,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,4,6}*960
Also Known As : {{20,4|2},{4,6|2}}. if this polytope has another name.
Group : SmallGroup(960,7401)
Rank : 4
Schlafli Type : {20,4,6}
Number of vertices, edges, etc : 20, 40, 12, 6
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {20,4,6,2} of size 1920
Vertex Figure Of :
   {2,20,4,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {20,2,6}*480, {10,4,6}*480
   3-fold quotients : {20,4,2}*320
   4-fold quotients : {20,2,3}*240, {10,2,6}*240
   5-fold quotients : {4,4,6}*192
   6-fold quotients : {20,2,2}*160, {10,4,2}*160
   8-fold quotients : {5,2,6}*120, {10,2,3}*120
   10-fold quotients : {2,4,6}*96a, {4,2,6}*96
   12-fold quotients : {10,2,2}*80
   15-fold quotients : {4,4,2}*64
   16-fold quotients : {5,2,3}*60
   20-fold quotients : {4,2,3}*48, {2,2,6}*48
   24-fold quotients : {5,2,2}*40
   30-fold quotients : {2,4,2}*32, {4,2,2}*32
   40-fold quotients : {2,2,3}*24
   60-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {20,4,12}*1920, {20,8,6}*1920a, {40,4,6}*1920a, {20,8,6}*1920b, {40,4,6}*1920b, {20,4,6}*1920a
Permutation Representation (GAP) :
s0 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)( 38, 39)
( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)( 58, 59)
( 61, 91)( 62, 95)( 63, 94)( 64, 93)( 65, 92)( 66, 96)( 67,100)( 68, 99)
( 69, 98)( 70, 97)( 71,101)( 72,105)( 73,104)( 74,103)( 75,102)( 76,106)
( 77,110)( 78,109)( 79,108)( 80,107)( 81,111)( 82,115)( 83,114)( 84,113)
( 85,112)( 86,116)( 87,120)( 88,119)( 89,118)( 90,117);;
s1 := (  1, 62)(  2, 61)(  3, 65)(  4, 64)(  5, 63)(  6, 67)(  7, 66)(  8, 70)
(  9, 69)( 10, 68)( 11, 72)( 12, 71)( 13, 75)( 14, 74)( 15, 73)( 16, 77)
( 17, 76)( 18, 80)( 19, 79)( 20, 78)( 21, 82)( 22, 81)( 23, 85)( 24, 84)
( 25, 83)( 26, 87)( 27, 86)( 28, 90)( 29, 89)( 30, 88)( 31, 92)( 32, 91)
( 33, 95)( 34, 94)( 35, 93)( 36, 97)( 37, 96)( 38,100)( 39, 99)( 40, 98)
( 41,102)( 42,101)( 43,105)( 44,104)( 45,103)( 46,107)( 47,106)( 48,110)
( 49,109)( 50,108)( 51,112)( 52,111)( 53,115)( 54,114)( 55,113)( 56,117)
( 57,116)( 58,120)( 59,119)( 60,118);;
s2 := (  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 21, 26)( 22, 27)( 23, 28)
( 24, 29)( 25, 30)( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 51, 56)
( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 61, 76)( 62, 77)( 63, 78)( 64, 79)
( 65, 80)( 66, 86)( 67, 87)( 68, 88)( 69, 89)( 70, 90)( 71, 81)( 72, 82)
( 73, 83)( 74, 84)( 75, 85)( 91,106)( 92,107)( 93,108)( 94,109)( 95,110)
( 96,116)( 97,117)( 98,118)( 99,119)(100,120)(101,111)(102,112)(103,113)
(104,114)(105,115);;
s3 := (  1,  6)(  2,  7)(  3,  8)(  4,  9)(  5, 10)( 16, 21)( 17, 22)( 18, 23)
( 19, 24)( 20, 25)( 31, 36)( 32, 37)( 33, 38)( 34, 39)( 35, 40)( 46, 51)
( 47, 52)( 48, 53)( 49, 54)( 50, 55)( 61, 66)( 62, 67)( 63, 68)( 64, 69)
( 65, 70)( 76, 81)( 77, 82)( 78, 83)( 79, 84)( 80, 85)( 91, 96)( 92, 97)
( 93, 98)( 94, 99)( 95,100)(106,111)(107,112)(108,113)(109,114)(110,115);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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(120)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)
( 38, 39)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)
( 58, 59)( 61, 91)( 62, 95)( 63, 94)( 64, 93)( 65, 92)( 66, 96)( 67,100)
( 68, 99)( 69, 98)( 70, 97)( 71,101)( 72,105)( 73,104)( 74,103)( 75,102)
( 76,106)( 77,110)( 78,109)( 79,108)( 80,107)( 81,111)( 82,115)( 83,114)
( 84,113)( 85,112)( 86,116)( 87,120)( 88,119)( 89,118)( 90,117);
s1 := Sym(120)!(  1, 62)(  2, 61)(  3, 65)(  4, 64)(  5, 63)(  6, 67)(  7, 66)
(  8, 70)(  9, 69)( 10, 68)( 11, 72)( 12, 71)( 13, 75)( 14, 74)( 15, 73)
( 16, 77)( 17, 76)( 18, 80)( 19, 79)( 20, 78)( 21, 82)( 22, 81)( 23, 85)
( 24, 84)( 25, 83)( 26, 87)( 27, 86)( 28, 90)( 29, 89)( 30, 88)( 31, 92)
( 32, 91)( 33, 95)( 34, 94)( 35, 93)( 36, 97)( 37, 96)( 38,100)( 39, 99)
( 40, 98)( 41,102)( 42,101)( 43,105)( 44,104)( 45,103)( 46,107)( 47,106)
( 48,110)( 49,109)( 50,108)( 51,112)( 52,111)( 53,115)( 54,114)( 55,113)
( 56,117)( 57,116)( 58,120)( 59,119)( 60,118);
s2 := Sym(120)!(  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 21, 26)( 22, 27)
( 23, 28)( 24, 29)( 25, 30)( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)
( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 61, 76)( 62, 77)( 63, 78)
( 64, 79)( 65, 80)( 66, 86)( 67, 87)( 68, 88)( 69, 89)( 70, 90)( 71, 81)
( 72, 82)( 73, 83)( 74, 84)( 75, 85)( 91,106)( 92,107)( 93,108)( 94,109)
( 95,110)( 96,116)( 97,117)( 98,118)( 99,119)(100,120)(101,111)(102,112)
(103,113)(104,114)(105,115);
s3 := Sym(120)!(  1,  6)(  2,  7)(  3,  8)(  4,  9)(  5, 10)( 16, 21)( 17, 22)
( 18, 23)( 19, 24)( 20, 25)( 31, 36)( 32, 37)( 33, 38)( 34, 39)( 35, 40)
( 46, 51)( 47, 52)( 48, 53)( 49, 54)( 50, 55)( 61, 66)( 62, 67)( 63, 68)
( 64, 69)( 65, 70)( 76, 81)( 77, 82)( 78, 83)( 79, 84)( 80, 85)( 91, 96)
( 92, 97)( 93, 98)( 94, 99)( 95,100)(106,111)(107,112)(108,113)(109,114)
(110,115);
poly := sub<Sym(120)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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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