Polytope of Type {10,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,6}*1500b
if this polytope has a name.
Group : SmallGroup(1500,125)
Rank : 3
Schlafli Type : {10,6}
Number of vertices, edges, etc : 125, 375, 75
Order of s0s1s2 : 15
Order of s0s1s2s1 : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   5-fold quotients : {10,6}*300
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 5.
      15 facets:
         15 of {10}*20
      25 vertex figures:
         25 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 5.
      15 facets:
         15 of {10}*20
      25 vertex figures:
         25 of {6}*12
   P/N, where N=<s0*s1*s0*s1> of order 5.
      35 facets:
         25 of {2}*4
         10 of {10}*20
      25 vertex figures:
         25 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 5.
      15 facets:
         15 of {10}*20
      25 vertex figures:
         25 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 5.
      15 facets:
         15 of {10}*20
      25 vertex figures:
         25 of {6}*12

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, 27)(  2, 26)(  3, 30)(  4, 29)(  5, 28)(  6, 33)(  7, 32)(  8, 31)(  9, 35)( 10, 34)( 11, 39)( 12, 38)( 13, 37)( 14, 36)( 15, 40)( 16, 45)( 17, 44)( 18, 43)( 19, 42)( 20, 41)( 21, 46)( 22, 50)( 23, 49)( 24, 48)( 25, 47)( 51,102)( 52,101)( 53,105)( 54,104)( 55,103)( 56,108)( 57,107)( 58,106)( 59,110)( 60,109)( 61,114)( 62,113)( 63,112)( 64,111)( 65,115)( 66,120)( 67,119)( 68,118)( 69,117)( 70,116)( 71,121)( 72,125)( 73,124)( 74,123)( 75,122)( 76, 77)( 78, 80)( 81, 83)( 84, 85)( 86, 89)( 87, 88)( 91, 95)( 92, 94)( 97,100)( 98, 99);;
s2 := (  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]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0 ];;
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, 27)(  2, 26)(  3, 30)(  4, 29)(  5, 28)(  6, 33)(  7, 32)(  8, 31)(  9, 35)( 10, 34)( 11, 39)( 12, 38)( 13, 37)( 14, 36)( 15, 40)( 16, 45)( 17, 44)( 18, 43)( 19, 42)( 20, 41)( 21, 46)( 22, 50)( 23, 49)( 24, 48)( 25, 47)( 51,102)( 52,101)( 53,105)( 54,104)( 55,103)( 56,108)( 57,107)( 58,106)( 59,110)( 60,109)( 61,114)( 62,113)( 63,112)( 64,111)( 65,115)( 66,120)( 67,119)( 68,118)( 69,117)( 70,116)( 71,121)( 72,125)( 73,124)( 74,123)( 75,122)( 76, 77)( 78, 80)( 81, 83)( 84, 85)( 86, 89)( 87, 88)( 91, 95)( 92, 94)( 97,100)( 98, 99);
s2 := 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>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0 >;
References : None.
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