Polytope of Type {8,6,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,10}*960
Also Known As : {{8,6|2},{6,10|2}}. if this polytope has another name.
Group : SmallGroup(960,8239)
Rank : 4
Schlafli Type : {8,6,10}
Number of vertices, edges, etc : 8, 24, 30, 10
Order of s₀s₁s₂s₃ : 120
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,6,10,2} of size 1920
Vertex Figure Of :
   {2,8,6,10} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6,10}*480a
   3-fold quotients : {8,2,10}*320
   4-fold quotients : {2,6,10}*240
   5-fold quotients : {8,6,2}*192
   6-fold quotients : {8,2,5}*160, {4,2,10}*160
   10-fold quotients : {4,6,2}*96a
   12-fold quotients : {4,2,5}*80, {2,2,10}*80
   15-fold quotients : {8,2,2}*64
   20-fold quotients : {2,6,2}*48
   24-fold quotients : {2,2,5}*40
   30-fold quotients : {4,2,2}*32
   40-fold quotients : {2,3,2}*24
   60-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,12,10}*1920a, {8,6,20}*1920, {16,6,10}*1920
Permutation Representation (GAP) :

s0 := ( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)( 36, 51)( 37, 52)( 38, 53)
( 39, 54)( 40, 55)( 41, 56)( 42, 57)( 43, 58)( 44, 59)( 45, 60)( 61, 91)
( 62, 92)( 63, 93)( 64, 94)( 65, 95)( 66, 96)( 67, 97)( 68, 98)( 69, 99)
( 70,100)( 71,101)( 72,102)( 73,103)( 74,104)( 75,105)( 76,106)( 77,107)
( 78,108)( 79,109)( 80,110)( 81,111)( 82,112)( 83,113)( 84,114)( 85,115)
( 86,116)( 87,117)( 88,118)( 89,119)( 90,120);;
s1 := (  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);;
s2 := (  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);;
s3 := (  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, 32)( 33, 35)( 36, 37)( 38, 40)
( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)( 58, 60)
( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)( 78, 80)
( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91, 92)( 93, 95)( 96, 97)( 98,100)
(101,102)(103,105)(106,107)(108,110)(111,112)(113,115)(116,117)(118,120);;
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, 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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(120)!( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)( 36, 51)( 37, 52)
( 38, 53)( 39, 54)( 40, 55)( 41, 56)( 42, 57)( 43, 58)( 44, 59)( 45, 60)
( 61, 91)( 62, 92)( 63, 93)( 64, 94)( 65, 95)( 66, 96)( 67, 97)( 68, 98)
( 69, 99)( 70,100)( 71,101)( 72,102)( 73,103)( 74,104)( 75,105)( 76,106)
( 77,107)( 78,108)( 79,109)( 80,110)( 81,111)( 82,112)( 83,113)( 84,114)
( 85,115)( 86,116)( 87,117)( 88,118)( 89,119)( 90,120);
s1 := 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);
s2 := 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);
s3 := 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, 32)( 33, 35)( 36, 37)
( 38, 40)( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)
( 58, 60)( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)
( 78, 80)( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91, 92)( 93, 95)( 96, 97)
( 98,100)(101,102)(103,105)(106,107)(108,110)(111,112)(113,115)(116,117)
(118,120);
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, 
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
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