Polytope of Type {5,10,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,10,3}*1500
if this polytope has a name.
Group : SmallGroup(1500,125)
Rank : 4
Schlafli Type : {5,10,3}
Number of vertices, edges, etc : 5, 125, 75, 15
Order of s₀s₁s₂s₃ : 15
Order of s₀s₁s₂s₃s₂s₁ : 10
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   25-fold quotients : {5,2,3}*60
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  5)(  3,  4)(  6, 21)(  7, 25)(  8, 24)(  9, 23)( 10, 22)( 11, 16)
( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 26,101)( 27,105)( 28,104)( 29,103)
( 30,102)( 31,121)( 32,125)( 33,124)( 34,123)( 35,122)( 36,116)( 37,120)
( 38,119)( 39,118)( 40,117)( 41,111)( 42,115)( 43,114)( 44,113)( 45,112)
( 46,106)( 47,110)( 48,109)( 49,108)( 50,107)( 51, 76)( 52, 80)( 53, 79)
( 54, 78)( 55, 77)( 56, 96)( 57,100)( 58, 99)( 59, 98)( 60, 97)( 61, 91)
( 62, 95)( 63, 94)( 64, 93)( 65, 92)( 66, 86)( 67, 90)( 68, 89)( 69, 88)
( 70, 87)( 71, 81)( 72, 85)( 73, 84)( 74, 83)( 75, 82);;
s1 := (  1, 31)(  2, 35)(  3, 34)(  4, 33)(  5, 32)(  6, 26)(  7, 30)(  8, 29)
(  9, 28)( 10, 27)( 11, 46)( 12, 50)( 13, 49)( 14, 48)( 15, 47)( 16, 41)
( 17, 45)( 18, 44)( 19, 43)( 20, 42)( 21, 36)( 22, 40)( 23, 39)( 24, 38)
( 25, 37)( 51,106)( 52,110)( 53,109)( 54,108)( 55,107)( 56,101)( 57,105)
( 58,104)( 59,103)( 60,102)( 61,121)( 62,125)( 63,124)( 64,123)( 65,122)
( 66,116)( 67,120)( 68,119)( 69,118)( 70,117)( 71,111)( 72,115)( 73,114)
( 74,113)( 75,112)( 76, 81)( 77, 85)( 78, 84)( 79, 83)( 80, 82)( 86, 96)
( 87,100)( 88, 99)( 89, 98)( 90, 97)( 92, 95)( 93, 94);;
s2 := (  6, 25)(  7, 21)(  8, 22)(  9, 23)( 10, 24)( 11, 19)( 12, 20)( 13, 16)
( 14, 17)( 15, 18)( 31, 50)( 32, 46)( 33, 47)( 34, 48)( 35, 49)( 36, 44)
( 37, 45)( 38, 41)( 39, 42)( 40, 43)( 56, 75)( 57, 71)( 58, 72)( 59, 73)
( 60, 74)( 61, 69)( 62, 70)( 63, 66)( 64, 67)( 65, 68)( 81,100)( 82, 96)
( 83, 97)( 84, 98)( 85, 99)( 86, 94)( 87, 95)( 88, 91)( 89, 92)( 90, 93)
(106,125)(107,121)(108,122)(109,123)(110,124)(111,119)(112,120)(113,116)
(114,117)(115,118);;
s3 := (  2, 25)(  3, 19)(  4, 13)(  5,  7)(  8, 24)(  9, 18)( 10, 12)( 14, 23)
( 15, 17)( 20, 22)( 27, 50)( 28, 44)( 29, 38)( 30, 32)( 33, 49)( 34, 43)
( 35, 37)( 39, 48)( 40, 42)( 45, 47)( 52, 75)( 53, 69)( 54, 63)( 55, 57)
( 58, 74)( 59, 68)( 60, 62)( 64, 73)( 65, 67)( 70, 72)( 77,100)( 78, 94)
( 79, 88)( 80, 82)( 83, 99)( 84, 93)( 85, 87)( 89, 98)( 90, 92)( 95, 97)
(102,125)(103,119)(104,113)(105,107)(108,124)(109,118)(110,112)(114,123)
(115,117)(120,122);;
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, s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(125)!(  2,  5)(  3,  4)(  6, 21)(  7, 25)(  8, 24)(  9, 23)( 10, 22)
( 11, 16)( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 26,101)( 27,105)( 28,104)
( 29,103)( 30,102)( 31,121)( 32,125)( 33,124)( 34,123)( 35,122)( 36,116)
( 37,120)( 38,119)( 39,118)( 40,117)( 41,111)( 42,115)( 43,114)( 44,113)
( 45,112)( 46,106)( 47,110)( 48,109)( 49,108)( 50,107)( 51, 76)( 52, 80)
( 53, 79)( 54, 78)( 55, 77)( 56, 96)( 57,100)( 58, 99)( 59, 98)( 60, 97)
( 61, 91)( 62, 95)( 63, 94)( 64, 93)( 65, 92)( 66, 86)( 67, 90)( 68, 89)
( 69, 88)( 70, 87)( 71, 81)( 72, 85)( 73, 84)( 74, 83)( 75, 82);
s1 := Sym(125)!(  1, 31)(  2, 35)(  3, 34)(  4, 33)(  5, 32)(  6, 26)(  7, 30)
(  8, 29)(  9, 28)( 10, 27)( 11, 46)( 12, 50)( 13, 49)( 14, 48)( 15, 47)
( 16, 41)( 17, 45)( 18, 44)( 19, 43)( 20, 42)( 21, 36)( 22, 40)( 23, 39)
( 24, 38)( 25, 37)( 51,106)( 52,110)( 53,109)( 54,108)( 55,107)( 56,101)
( 57,105)( 58,104)( 59,103)( 60,102)( 61,121)( 62,125)( 63,124)( 64,123)
( 65,122)( 66,116)( 67,120)( 68,119)( 69,118)( 70,117)( 71,111)( 72,115)
( 73,114)( 74,113)( 75,112)( 76, 81)( 77, 85)( 78, 84)( 79, 83)( 80, 82)
( 86, 96)( 87,100)( 88, 99)( 89, 98)( 90, 97)( 92, 95)( 93, 94);
s2 := Sym(125)!(  6, 25)(  7, 21)(  8, 22)(  9, 23)( 10, 24)( 11, 19)( 12, 20)
( 13, 16)( 14, 17)( 15, 18)( 31, 50)( 32, 46)( 33, 47)( 34, 48)( 35, 49)
( 36, 44)( 37, 45)( 38, 41)( 39, 42)( 40, 43)( 56, 75)( 57, 71)( 58, 72)
( 59, 73)( 60, 74)( 61, 69)( 62, 70)( 63, 66)( 64, 67)( 65, 68)( 81,100)
( 82, 96)( 83, 97)( 84, 98)( 85, 99)( 86, 94)( 87, 95)( 88, 91)( 89, 92)
( 90, 93)(106,125)(107,121)(108,122)(109,123)(110,124)(111,119)(112,120)
(113,116)(114,117)(115,118);
s3 := Sym(125)!(  2, 25)(  3, 19)(  4, 13)(  5,  7)(  8, 24)(  9, 18)( 10, 12)
( 14, 23)( 15, 17)( 20, 22)( 27, 50)( 28, 44)( 29, 38)( 30, 32)( 33, 49)
( 34, 43)( 35, 37)( 39, 48)( 40, 42)( 45, 47)( 52, 75)( 53, 69)( 54, 63)
( 55, 57)( 58, 74)( 59, 68)( 60, 62)( 64, 73)( 65, 67)( 70, 72)( 77,100)
( 78, 94)( 79, 88)( 80, 82)( 83, 99)( 84, 93)( 85, 87)( 89, 98)( 90, 92)
( 95, 97)(102,125)(103,119)(104,113)(105,107)(108,124)(109,118)(110,112)
(114,123)(115,117)(120,122);
poly := sub<Sym(125)|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, 
s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
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