Polytope of Type {10,4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,4,12}*960
Also Known As : {{10,4|2},{4,12|2}}. if this polytope has another name.
Group : SmallGroup(960,7400)
Rank : 4
Schlafli Type : {10,4,12}
Number of vertices, edges, etc : 10, 20, 24, 12
Order of s₀s₁s₂s₃ : 60
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {10,4,12,2} of size 1920
Vertex Figure Of :
   {2,10,4,12} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {10,2,12}*480, {10,4,6}*480
   3-fold quotients : {10,4,4}*320
   4-fold quotients : {5,2,12}*240, {10,2,6}*240
   5-fold quotients : {2,4,12}*192a
   6-fold quotients : {10,2,4}*160, {10,4,2}*160
   8-fold quotients : {5,2,6}*120, {10,2,3}*120
   10-fold quotients : {2,2,12}*96, {2,4,6}*96a
   12-fold quotients : {5,2,4}*80, {10,2,2}*80
   15-fold quotients : {2,4,4}*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 : {2,2,4}*32, {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 : {20,4,12}*1920, {10,8,12}*1920a, {10,4,24}*1920a, {10,8,12}*1920b, {10,4,24}*1920b, {10,4,12}*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,  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, 77)( 62, 76)( 63, 80)( 64, 79)( 65, 78)( 66, 82)( 67, 81)( 68, 85)
( 69, 84)( 70, 83)( 71, 87)( 72, 86)( 73, 90)( 74, 89)( 75, 88)( 91,107)
( 92,106)( 93,110)( 94,109)( 95,108)( 96,112)( 97,111)( 98,115)( 99,114)
(100,113)(101,117)(102,116)(103,120)(104,119)(105,118);;
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, 91)( 32, 92)
( 33, 93)( 34, 94)( 35, 95)( 36,101)( 37,102)( 38,103)( 39,104)( 40,105)
( 41, 96)( 42, 97)( 43, 98)( 44, 99)( 45,100)( 46,106)( 47,107)( 48,108)
( 49,109)( 50,110)( 51,116)( 52,117)( 53,118)( 54,119)( 55,120)( 56,111)
( 57,112)( 58,113)( 59,114)( 60,115);;
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, 96)( 62, 97)( 63, 98)( 64, 99)
( 65,100)( 66, 91)( 67, 92)( 68, 93)( 69, 94)( 70, 95)( 71,101)( 72,102)
( 73,103)( 74,104)( 75,105)( 76,111)( 77,112)( 78,113)( 79,114)( 80,115)
( 81,106)( 82,107)( 83,108)( 84,109)( 85,110)( 86,116)( 87,117)( 88,118)
( 89,119)( 90,120);;
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*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s0*s1*s0*s1*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*s2*s3*s2*s3 ];;
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, 32)( 33, 35)( 36, 37)
( 38, 40)( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)
( 58, 60)( 61, 77)( 62, 76)( 63, 80)( 64, 79)( 65, 78)( 66, 82)( 67, 81)
( 68, 85)( 69, 84)( 70, 83)( 71, 87)( 72, 86)( 73, 90)( 74, 89)( 75, 88)
( 91,107)( 92,106)( 93,110)( 94,109)( 95,108)( 96,112)( 97,111)( 98,115)
( 99,114)(100,113)(101,117)(102,116)(103,120)(104,119)(105,118);
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, 91)
( 32, 92)( 33, 93)( 34, 94)( 35, 95)( 36,101)( 37,102)( 38,103)( 39,104)
( 40,105)( 41, 96)( 42, 97)( 43, 98)( 44, 99)( 45,100)( 46,106)( 47,107)
( 48,108)( 49,109)( 50,110)( 51,116)( 52,117)( 53,118)( 54,119)( 55,120)
( 56,111)( 57,112)( 58,113)( 59,114)( 60,115);
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, 96)( 62, 97)( 63, 98)
( 64, 99)( 65,100)( 66, 91)( 67, 92)( 68, 93)( 69, 94)( 70, 95)( 71,101)
( 72,102)( 73,103)( 74,104)( 75,105)( 76,111)( 77,112)( 78,113)( 79,114)
( 80,115)( 81,106)( 82,107)( 83,108)( 84,109)( 85,110)( 86,116)( 87,117)
( 88,118)( 89,119)( 90,120);
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*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*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*s2*s3*s2*s3 >;

References : None.
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