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Polytope of Type {10,8,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,8,6}*960
Also Known As : {{10,8|2},{8,6|2}}. if this polytope has another name.
Group : SmallGroup(960,8239)
Rank : 4
Schlafli Type : {10,8,6}
Number of vertices, edges, etc : 10, 40, 24, 6
Order of s0s1s2s3 : 120
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {10,8,6,2} of size 1920
Vertex Figure Of :
   {2,10,8,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {10,4,6}*480
   3-fold quotients : {10,8,2}*320
   4-fold quotients : {10,2,6}*240
   5-fold quotients : {2,8,6}*192
   6-fold quotients : {10,4,2}*160
   8-fold quotients : {5,2,6}*120, {10,2,3}*120
   10-fold quotients : {2,4,6}*96a
   12-fold quotients : {10,2,2}*80
   15-fold quotients : {2,8,2}*64
   16-fold quotients : {5,2,3}*60
   20-fold quotients : {2,2,6}*48
   24-fold quotients : {5,2,2}*40
   30-fold quotients : {2,4,2}*32
   40-fold quotients : {2,2,3}*24
   60-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {10,8,12}*1920a, {20,8,6}*1920a, {10,16,6}*1920
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)
( 62, 65)( 63, 64)( 67, 70)( 68, 69)( 72, 75)( 73, 74)( 77, 80)( 78, 79)
( 82, 85)( 83, 84)( 87, 90)( 88, 89)( 92, 95)( 93, 94)( 97,100)( 98, 99)
(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119);;
s1 := (  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);;
s2 := (  1, 61)(  2, 62)(  3, 63)(  4, 64)(  5, 65)(  6, 71)(  7, 72)(  8, 73)
(  9, 74)( 10, 75)( 11, 66)( 12, 67)( 13, 68)( 14, 69)( 15, 70)( 16, 76)
( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 86)( 22, 87)( 23, 88)( 24, 89)
( 25, 90)( 26, 81)( 27, 82)( 28, 83)( 29, 84)( 30, 85)( 31,106)( 32,107)
( 33,108)( 34,109)( 35,110)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)
( 41,111)( 42,112)( 43,113)( 44,114)( 45,115)( 46, 91)( 47, 92)( 48, 93)
( 49, 94)( 50, 95)( 51,101)( 52,102)( 53,103)( 54,104)( 55,105)( 56, 96)
( 57, 97)( 58, 98)( 59, 99)( 60,100);;
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*s3*s2*s1*s2*s3*s2, 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, 
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)( 62, 65)( 63, 64)( 67, 70)( 68, 69)( 72, 75)( 73, 74)( 77, 80)
( 78, 79)( 82, 85)( 83, 84)( 87, 90)( 88, 89)( 92, 95)( 93, 94)( 97,100)
( 98, 99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)
(118,119);
s1 := 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);
s2 := Sym(120)!(  1, 61)(  2, 62)(  3, 63)(  4, 64)(  5, 65)(  6, 71)(  7, 72)
(  8, 73)(  9, 74)( 10, 75)( 11, 66)( 12, 67)( 13, 68)( 14, 69)( 15, 70)
( 16, 76)( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 86)( 22, 87)( 23, 88)
( 24, 89)( 25, 90)( 26, 81)( 27, 82)( 28, 83)( 29, 84)( 30, 85)( 31,106)
( 32,107)( 33,108)( 34,109)( 35,110)( 36,116)( 37,117)( 38,118)( 39,119)
( 40,120)( 41,111)( 42,112)( 43,113)( 44,114)( 45,115)( 46, 91)( 47, 92)
( 48, 93)( 49, 94)( 50, 95)( 51,101)( 52,102)( 53,103)( 54,104)( 55,105)
( 56, 96)( 57, 97)( 58, 98)( 59, 99)( 60,100);
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*s3*s2*s1*s2*s3*s2, 
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, 
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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