Part of the Atlas of Small Regular Polytopes

Polytope of Type {10,15}

Atlas Canonical Name {10,15}*1500d

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Overview

Group
SmallGroup(1500,37)
Rank
3
Schläfli Type
{10,15}
Vertices, edges, …
50, 375, 75
Order of s0s1s2
6
Order of s0s1s2s1
10
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

5-fold

125-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1)^4*s2*(s1*s0)^2*s2> of order 5

15 facets

10 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle