Polytope of Type {10,4,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,4,10}*800
Also Known As : {{10,4|2},{4,10|2}}. if this polytope has another name.
Group : SmallGroup(800,1134)
Rank : 4
Schlafli Type : {10,4,10}
Number of vertices, edges, etc : 10, 20, 20, 10
Order of s₀s₁s₂s₃ : 20
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {10,4,10,2} of size 1600
Vertex Figure Of :
   {2,10,4,10} of size 1600
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {10,2,10}*400
   4-fold quotients : {5,2,10}*200, {10,2,5}*200
   5-fold quotients : {2,4,10}*160, {10,4,2}*160
   8-fold quotients : {5,2,5}*100
   10-fold quotients : {2,2,10}*80, {10,2,2}*80
   20-fold quotients : {2,2,5}*40, {5,2,2}*40
   25-fold quotients : {2,4,2}*32
   50-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {10,4,20}*1600, {20,4,10}*1600, {10,8,10}*1600
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);;
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, 77)( 52, 76)( 53, 80)( 54, 79)
( 55, 78)( 56, 82)( 57, 81)( 58, 85)( 59, 84)( 60, 83)( 61, 87)( 62, 86)
( 63, 90)( 64, 89)( 65, 88)( 66, 92)( 67, 91)( 68, 95)( 69, 94)( 70, 93)
( 71, 97)( 72, 96)( 73,100)( 74, 99)( 75, 98);;
s2 := (  1, 51)(  2, 52)(  3, 53)(  4, 54)(  5, 55)(  6, 71)(  7, 72)(  8, 73)
(  9, 74)( 10, 75)( 11, 66)( 12, 67)( 13, 68)( 14, 69)( 15, 70)( 16, 61)
( 17, 62)( 18, 63)( 19, 64)( 20, 65)( 21, 56)( 22, 57)( 23, 58)( 24, 59)
( 25, 60)( 26, 76)( 27, 77)( 28, 78)( 29, 79)( 30, 80)( 31, 96)( 32, 97)
( 33, 98)( 34, 99)( 35,100)( 36, 91)( 37, 92)( 38, 93)( 39, 94)( 40, 95)
( 41, 86)( 42, 87)( 43, 88)( 44, 89)( 45, 90)( 46, 81)( 47, 82)( 48, 83)
( 49, 84)( 50, 85);;
s3 := (  1,  6)(  2,  7)(  3,  8)(  4,  9)(  5, 10)( 11, 21)( 12, 22)( 13, 23)
( 14, 24)( 15, 25)( 26, 31)( 27, 32)( 28, 33)( 29, 34)( 30, 35)( 36, 46)
( 37, 47)( 38, 48)( 39, 49)( 40, 50)( 51, 56)( 52, 57)( 53, 58)( 54, 59)
( 55, 60)( 61, 71)( 62, 72)( 63, 73)( 64, 74)( 65, 75)( 76, 81)( 77, 82)
( 78, 83)( 79, 84)( 80, 85)( 86, 96)( 87, 97)( 88, 98)( 89, 99)( 90,100);;
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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(100)!(  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);
s1 := Sym(100)!(  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, 77)( 52, 76)( 53, 80)
( 54, 79)( 55, 78)( 56, 82)( 57, 81)( 58, 85)( 59, 84)( 60, 83)( 61, 87)
( 62, 86)( 63, 90)( 64, 89)( 65, 88)( 66, 92)( 67, 91)( 68, 95)( 69, 94)
( 70, 93)( 71, 97)( 72, 96)( 73,100)( 74, 99)( 75, 98);
s2 := Sym(100)!(  1, 51)(  2, 52)(  3, 53)(  4, 54)(  5, 55)(  6, 71)(  7, 72)
(  8, 73)(  9, 74)( 10, 75)( 11, 66)( 12, 67)( 13, 68)( 14, 69)( 15, 70)
( 16, 61)( 17, 62)( 18, 63)( 19, 64)( 20, 65)( 21, 56)( 22, 57)( 23, 58)
( 24, 59)( 25, 60)( 26, 76)( 27, 77)( 28, 78)( 29, 79)( 30, 80)( 31, 96)
( 32, 97)( 33, 98)( 34, 99)( 35,100)( 36, 91)( 37, 92)( 38, 93)( 39, 94)
( 40, 95)( 41, 86)( 42, 87)( 43, 88)( 44, 89)( 45, 90)( 46, 81)( 47, 82)
( 48, 83)( 49, 84)( 50, 85);
s3 := Sym(100)!(  1,  6)(  2,  7)(  3,  8)(  4,  9)(  5, 10)( 11, 21)( 12, 22)
( 13, 23)( 14, 24)( 15, 25)( 26, 31)( 27, 32)( 28, 33)( 29, 34)( 30, 35)
( 36, 46)( 37, 47)( 38, 48)( 39, 49)( 40, 50)( 51, 56)( 52, 57)( 53, 58)
( 54, 59)( 55, 60)( 61, 71)( 62, 72)( 63, 73)( 64, 74)( 65, 75)( 76, 81)
( 77, 82)( 78, 83)( 79, 84)( 80, 85)( 86, 96)( 87, 97)( 88, 98)( 89, 99)
( 90,100);
poly := sub<Sym(100)|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 >;

References : None.
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