Polytope of Type {3,10,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,10,5}*1500
if this polytope has a name.
Group : SmallGroup(1500,125)
Rank : 4
Schlafli Type : {3,10,5}
Number of vertices, edges, etc : 15, 75, 125, 5
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 : {3,2,5}*60
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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