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Polytope of Type {6,10,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,10,8}*960
Also Known As : {{6,10|2},{10,8|2}}. if this polytope has another name.
Group : SmallGroup(960,8239)
Rank : 4
Schlafli Type : {6,10,8}
Number of vertices, edges, etc : 6, 30, 40, 8
Order of s0s1s2s3 : 120
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,10,8,2} of size 1920
Vertex Figure Of :
   {2,6,10,8} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,10,4}*480
   3-fold quotients : {2,10,8}*320
   4-fold quotients : {6,10,2}*240
   5-fold quotients : {6,2,8}*192
   6-fold quotients : {2,10,4}*160
   10-fold quotients : {3,2,8}*96, {6,2,4}*96
   12-fold quotients : {2,10,2}*80
   15-fold quotients : {2,2,8}*64
   20-fold quotients : {3,2,4}*48, {6,2,2}*48
   24-fold quotients : {2,5,2}*40
   30-fold quotients : {2,2,4}*32
   40-fold quotients : {3,2,2}*24
   60-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,20,8}*1920a, {12,10,8}*1920, {6,10,16}*1920
Permutation Representation (GAP) :
s0 := (  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)( 66, 71)( 67, 72)( 68, 73)( 69, 74)
( 70, 75)( 81, 86)( 82, 87)( 83, 88)( 84, 89)( 85, 90)( 96,101)( 97,102)
( 98,103)( 99,104)(100,105)(111,116)(112,117)(113,118)(114,119)(115,120);;
s1 := (  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 12, 15)( 13, 14)( 16, 21)
( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 30)( 28, 29)( 31, 36)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 42, 45)( 43, 44)( 46, 51)( 47, 55)( 48, 54)
( 49, 53)( 50, 52)( 57, 60)( 58, 59)( 61, 66)( 62, 70)( 63, 69)( 64, 68)
( 65, 67)( 72, 75)( 73, 74)( 76, 81)( 77, 85)( 78, 84)( 79, 83)( 80, 82)
( 87, 90)( 88, 89)( 91, 96)( 92,100)( 93, 99)( 94, 98)( 95, 97)(102,105)
(103,104)(106,111)(107,115)(108,114)(109,113)(110,112)(117,120)(118,119);;
s2 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)( 18, 20)
( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 47)( 32, 46)( 33, 50)( 34, 49)
( 35, 48)( 36, 52)( 37, 51)( 38, 55)( 39, 54)( 40, 53)( 41, 57)( 42, 56)
( 43, 60)( 44, 59)( 45, 58)( 61, 92)( 62, 91)( 63, 95)( 64, 94)( 65, 93)
( 66, 97)( 67, 96)( 68,100)( 69, 99)( 70, 98)( 71,102)( 72,101)( 73,105)
( 74,104)( 75,103)( 76,107)( 77,106)( 78,110)( 79,109)( 80,108)( 81,112)
( 82,111)( 83,115)( 84,114)( 85,113)( 86,117)( 87,116)( 88,120)( 89,119)
( 90,118);;
s3 := (  1, 61)(  2, 62)(  3, 63)(  4, 64)(  5, 65)(  6, 66)(  7, 67)(  8, 68)
(  9, 69)( 10, 70)( 11, 71)( 12, 72)( 13, 73)( 14, 74)( 15, 75)( 16, 76)
( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 81)( 22, 82)( 23, 83)( 24, 84)
( 25, 85)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31,106)( 32,107)
( 33,108)( 34,109)( 35,110)( 36,111)( 37,112)( 38,113)( 39,114)( 40,115)
( 41,116)( 42,117)( 43,118)( 44,119)( 45,120)( 46, 91)( 47, 92)( 48, 93)
( 49, 94)( 50, 95)( 51, 96)( 52, 97)( 53, 98)( 54, 99)( 55,100)( 56,101)
( 57,102)( 58,103)( 59,104)( 60,105);;
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*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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(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)( 66, 71)( 67, 72)( 68, 73)
( 69, 74)( 70, 75)( 81, 86)( 82, 87)( 83, 88)( 84, 89)( 85, 90)( 96,101)
( 97,102)( 98,103)( 99,104)(100,105)(111,116)(112,117)(113,118)(114,119)
(115,120);
s1 := Sym(120)!(  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 12, 15)( 13, 14)
( 16, 21)( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 30)( 28, 29)( 31, 36)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 42, 45)( 43, 44)( 46, 51)( 47, 55)
( 48, 54)( 49, 53)( 50, 52)( 57, 60)( 58, 59)( 61, 66)( 62, 70)( 63, 69)
( 64, 68)( 65, 67)( 72, 75)( 73, 74)( 76, 81)( 77, 85)( 78, 84)( 79, 83)
( 80, 82)( 87, 90)( 88, 89)( 91, 96)( 92,100)( 93, 99)( 94, 98)( 95, 97)
(102,105)(103,104)(106,111)(107,115)(108,114)(109,113)(110,112)(117,120)
(118,119);
s2 := Sym(120)!(  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)
( 18, 20)( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 47)( 32, 46)( 33, 50)
( 34, 49)( 35, 48)( 36, 52)( 37, 51)( 38, 55)( 39, 54)( 40, 53)( 41, 57)
( 42, 56)( 43, 60)( 44, 59)( 45, 58)( 61, 92)( 62, 91)( 63, 95)( 64, 94)
( 65, 93)( 66, 97)( 67, 96)( 68,100)( 69, 99)( 70, 98)( 71,102)( 72,101)
( 73,105)( 74,104)( 75,103)( 76,107)( 77,106)( 78,110)( 79,109)( 80,108)
( 81,112)( 82,111)( 83,115)( 84,114)( 85,113)( 86,117)( 87,116)( 88,120)
( 89,119)( 90,118);
s3 := Sym(120)!(  1, 61)(  2, 62)(  3, 63)(  4, 64)(  5, 65)(  6, 66)(  7, 67)
(  8, 68)(  9, 69)( 10, 70)( 11, 71)( 12, 72)( 13, 73)( 14, 74)( 15, 75)
( 16, 76)( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 81)( 22, 82)( 23, 83)
( 24, 84)( 25, 85)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31,106)
( 32,107)( 33,108)( 34,109)( 35,110)( 36,111)( 37,112)( 38,113)( 39,114)
( 40,115)( 41,116)( 42,117)( 43,118)( 44,119)( 45,120)( 46, 91)( 47, 92)
( 48, 93)( 49, 94)( 50, 95)( 51, 96)( 52, 97)( 53, 98)( 54, 99)( 55,100)
( 56,101)( 57,102)( 58,103)( 59,104)( 60,105);
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*s3*s2*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
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