Polytope of Type {10,12,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,12,4}*960a
Also Known As : {{10,12|2},{12,4|2}}. if this polytope has another name.
Group : SmallGroup(960,7400)
Rank : 4
Schlafli Type : {10,12,4}
Number of vertices, edges, etc : 10, 60, 24, 4
Order of s₀s₁s₂s₃ : 60
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {10,12,4,2} of size 1920
Vertex Figure Of :
   {2,10,12,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {10,12,2}*480, {10,6,4}*480a
   3-fold quotients : {10,4,4}*320
   4-fold quotients : {10,6,2}*240
   5-fold quotients : {2,12,4}*192a
   6-fold quotients : {10,2,4}*160, {10,4,2}*160
   10-fold quotients : {2,12,2}*96, {2,6,4}*96a
   12-fold quotients : {5,2,4}*80, {10,2,2}*80
   15-fold quotients : {2,4,4}*64
   20-fold quotients : {2,6,2}*48
   24-fold quotients : {5,2,2}*40
   30-fold quotients : {2,2,4}*32, {2,4,2}*32
   40-fold quotients : {2,3,2}*24
   60-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {20,12,4}*1920a, {10,12,8}*1920a, {10,24,4}*1920a, {10,12,8}*1920b, {10,24,4}*1920b, {10,12,4}*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)
( 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, 12)(  7, 11)(  8, 15)(  9, 14)( 10, 13)( 16, 17)
( 18, 20)( 21, 27)( 22, 26)( 23, 30)( 24, 29)( 25, 28)( 31, 32)( 33, 35)
( 36, 42)( 37, 41)( 38, 45)( 39, 44)( 40, 43)( 46, 47)( 48, 50)( 51, 57)
( 52, 56)( 53, 60)( 54, 59)( 55, 58)( 61, 92)( 62, 91)( 63, 95)( 64, 94)
( 65, 93)( 66,102)( 67,101)( 68,105)( 69,104)( 70,103)( 71, 97)( 72, 96)
( 73,100)( 74, 99)( 75, 98)( 76,107)( 77,106)( 78,110)( 79,109)( 80,108)
( 81,117)( 82,116)( 83,120)( 84,119)( 85,118)( 86,112)( 87,111)( 88,115)
( 89,114)( 90,113);;
s2 := (  1, 66)(  2, 67)(  3, 68)(  4, 69)(  5, 70)(  6, 61)(  7, 62)(  8, 63)
(  9, 64)( 10, 65)( 11, 71)( 12, 72)( 13, 73)( 14, 74)( 15, 75)( 16, 81)
( 17, 82)( 18, 83)( 19, 84)( 20, 85)( 21, 76)( 22, 77)( 23, 78)( 24, 79)
( 25, 80)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31, 96)( 32, 97)
( 33, 98)( 34, 99)( 35,100)( 36, 91)( 37, 92)( 38, 93)( 39, 94)( 40, 95)
( 41,101)( 42,102)( 43,103)( 44,104)( 45,105)( 46,111)( 47,112)( 48,113)
( 49,114)( 50,115)( 51,106)( 52,107)( 53,108)( 54,109)( 55,110)( 56,116)
( 57,117)( 58,118)( 59,119)( 60,120);;
s3 := ( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)
( 69, 84)( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)( 91,106)
( 92,107)( 93,108)( 94,109)( 95,110)( 96,111)( 97,112)( 98,113)( 99,114)
(100,115)(101,116)(102,117)(103,118)(104,119)(105,120);;
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, 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, 
s1*s2*s1*s2*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)!(  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, 12)(  7, 11)(  8, 15)(  9, 14)( 10, 13)
( 16, 17)( 18, 20)( 21, 27)( 22, 26)( 23, 30)( 24, 29)( 25, 28)( 31, 32)
( 33, 35)( 36, 42)( 37, 41)( 38, 45)( 39, 44)( 40, 43)( 46, 47)( 48, 50)
( 51, 57)( 52, 56)( 53, 60)( 54, 59)( 55, 58)( 61, 92)( 62, 91)( 63, 95)
( 64, 94)( 65, 93)( 66,102)( 67,101)( 68,105)( 69,104)( 70,103)( 71, 97)
( 72, 96)( 73,100)( 74, 99)( 75, 98)( 76,107)( 77,106)( 78,110)( 79,109)
( 80,108)( 81,117)( 82,116)( 83,120)( 84,119)( 85,118)( 86,112)( 87,111)
( 88,115)( 89,114)( 90,113);
s2 := Sym(120)!(  1, 66)(  2, 67)(  3, 68)(  4, 69)(  5, 70)(  6, 61)(  7, 62)
(  8, 63)(  9, 64)( 10, 65)( 11, 71)( 12, 72)( 13, 73)( 14, 74)( 15, 75)
( 16, 81)( 17, 82)( 18, 83)( 19, 84)( 20, 85)( 21, 76)( 22, 77)( 23, 78)
( 24, 79)( 25, 80)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31, 96)
( 32, 97)( 33, 98)( 34, 99)( 35,100)( 36, 91)( 37, 92)( 38, 93)( 39, 94)
( 40, 95)( 41,101)( 42,102)( 43,103)( 44,104)( 45,105)( 46,111)( 47,112)
( 48,113)( 49,114)( 50,115)( 51,106)( 52,107)( 53,108)( 54,109)( 55,110)
( 56,116)( 57,117)( 58,118)( 59,119)( 60,120);
s3 := Sym(120)!( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)
( 68, 83)( 69, 84)( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)
( 91,106)( 92,107)( 93,108)( 94,109)( 95,110)( 96,111)( 97,112)( 98,113)
( 99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120);
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, 
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, 
s1*s2*s1*s2*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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