Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,10}

Atlas Canonical Name {6,10}*1500b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1500,125)
Rank
3
Schläfli Type
{6,10}
Vertices, edges, …
75, 375, 125
Order of s0s1s2
15
Order of s0s1s2s1
10
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable

Quotients maximal quotients in bold

5-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*s0*(s2*s1)^3*s0*s2*s1*s2> of order 5

25 facets

15 vertex figures

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 5

25 facets

35 vertex figures

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

25 facets

15 vertex figures

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

25 facets

15 vertex figures

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

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

References

None.

to this polytope.

Twisty Puzzle