Part of the Atlas of Small Regular Polytopes

Polytope of Type {15,10}

Atlas Canonical Name {15,10}*1500g

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1500,126)
Rank
3
Schläfli Type
{15,10}
Vertices, edges, …
75, 375, 50
Order of s0s1s2
30
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

25-fold

75-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*s2*s1*s0*(s2*s1)^2*s2> of order 5

10 facets

15 vertex figures

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

10 facets

35 vertex figures

P/N, where N=<s0*s1*s0*s2*s1*s0*(s1*s2)^3> of order 5

10 facets

15 vertex figures

P/N, where N=<s0*s1*(s0*(s2*s1)^3)^2> of order 5

10 facets

15 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle