Part of the Atlas of Small Regular Polytopes

Polytope of Type {7,8}

Atlas Canonical Name {7,8}*1792b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1792,1083551)
Rank
3
Schläfli Type
{7,8}
Vertices, edges, …
112, 448, 128
Order of s0s1s2
7
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable
  • Self-Petrie

Quotients maximal quotients in bold

2-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=<(s1*s0)^2*s1*(s2*s1*s0)^2*s2*s1> of order 2

64 facets

56 vertex figures

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

64 facets

56 vertex figures

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

64 facets

56 vertex figures

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

32 facets

28 vertex figures

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

32 facets

28 vertex figures

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

32 facets

28 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*s1*s2*s1, (s0*s1)^3*s2*s1*s0*s2*(s1*s0)^2*s2> of order 4

32 facets

28 vertex figures

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

32 facets

28 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*s1*s2*s1, s0*s1*s2*(s1*s0)^2*s1*s2*s1*s0*s1*s2, (s1*s0)^2*s1*s2*(s1*s0)^2*s2*s1*s2> of order 8

16 facets

14 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  3, 17)(  4, 18)(  5, 97)(  6, 98)(  7,113)(  8,114)(  9, 33)( 10, 34)( 11, 49)( 12, 50)( 13, 65)( 14, 66)( 15, 81)( 16, 82)( 19, 20)( 21,100)( 22, 99)( 23,115)( 24,116)( 25, 36)( 26, 35)( 27, 51)( 28, 52)( 29, 67)( 30, 68)( 31, 84)( 32, 83)( 37,105)( 38,106)( 39,122)( 40,121)( 43, 58)( 44, 57)( 45, 73)( 46, 74)( 47, 90)( 48, 89)( 53,108)( 54,107)( 55,124)( 56,123)( 59, 60)( 61, 75)( 62, 76)( 63, 91)( 64, 92)( 69,110)( 70,109)( 71,126)( 72,125)( 77, 78)( 79, 94)( 80, 93)( 85,111)( 86,112)( 87,128)( 88,127)(101,102)(103,117)(104,118);;
s1 := (  3, 33)(  4, 34)(  5,113)(  6,114)(  7, 81)(  8, 82)(  9, 65)( 10, 66)( 11, 97)( 12, 98)( 13, 49)( 14, 50)( 15, 17)( 16, 18)( 19, 48)( 20, 47)( 21,128)( 22,127)( 23, 95)( 24, 96)( 25, 80)( 26, 79)( 27,111)( 28,112)( 29, 63)( 30, 64)( 31, 32)( 35, 36)( 37,115)( 38,116)( 39, 84)( 40, 83)( 41, 67)( 42, 68)( 43,100)( 44, 99)( 45, 51)( 46, 52)( 53,126)( 54,125)( 55, 94)( 56, 93)( 57, 78)( 58, 77)( 59,110)( 60,109)( 69,122)( 70,121)( 71, 90)( 72, 89)( 75,105)( 76,106)( 85,119)( 86,120)( 87, 88)( 91,103)( 92,104)(101,124)(102,123)(107,108);;
s2 := (  1, 42)(  2, 41)(  3, 43)(  4, 44)(  5, 46)(  6, 45)(  7, 47)(  8, 48)(  9, 34)( 10, 33)( 11, 35)( 12, 36)( 13, 38)( 14, 37)( 15, 39)( 16, 40)( 17, 58)( 18, 57)( 19, 59)( 20, 60)( 21, 62)( 22, 61)( 23, 63)( 24, 64)( 25, 50)( 26, 49)( 27, 51)( 28, 52)( 29, 54)( 30, 53)( 31, 55)( 32, 56)( 65,106)( 66,105)( 67,107)( 68,108)( 69,110)( 70,109)( 71,111)( 72,112)( 73, 98)( 74, 97)( 75, 99)( 76,100)( 77,102)( 78,101)( 79,103)( 80,104)( 81,122)( 82,121)( 83,123)( 84,124)( 85,126)( 86,125)( 87,127)( 88,128)( 89,114)( 90,113)( 91,115)( 92,116)( 93,118)( 94,117)( 95,119)( 96,120);;
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, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(128)!(  3, 17)(  4, 18)(  5, 97)(  6, 98)(  7,113)(  8,114)(  9, 33)( 10, 34)( 11, 49)( 12, 50)( 13, 65)( 14, 66)( 15, 81)( 16, 82)( 19, 20)( 21,100)( 22, 99)( 23,115)( 24,116)( 25, 36)( 26, 35)( 27, 51)( 28, 52)( 29, 67)( 30, 68)( 31, 84)( 32, 83)( 37,105)( 38,106)( 39,122)( 40,121)( 43, 58)( 44, 57)( 45, 73)( 46, 74)( 47, 90)( 48, 89)( 53,108)( 54,107)( 55,124)( 56,123)( 59, 60)( 61, 75)( 62, 76)( 63, 91)( 64, 92)( 69,110)( 70,109)( 71,126)( 72,125)( 77, 78)( 79, 94)( 80, 93)( 85,111)( 86,112)( 87,128)( 88,127)(101,102)(103,117)(104,118);
s1 := Sym(128)!(  3, 33)(  4, 34)(  5,113)(  6,114)(  7, 81)(  8, 82)(  9, 65)( 10, 66)( 11, 97)( 12, 98)( 13, 49)( 14, 50)( 15, 17)( 16, 18)( 19, 48)( 20, 47)( 21,128)( 22,127)( 23, 95)( 24, 96)( 25, 80)( 26, 79)( 27,111)( 28,112)( 29, 63)( 30, 64)( 31, 32)( 35, 36)( 37,115)( 38,116)( 39, 84)( 40, 83)( 41, 67)( 42, 68)( 43,100)( 44, 99)( 45, 51)( 46, 52)( 53,126)( 54,125)( 55, 94)( 56, 93)( 57, 78)( 58, 77)( 59,110)( 60,109)( 69,122)( 70,121)( 71, 90)( 72, 89)( 75,105)( 76,106)( 85,119)( 86,120)( 87, 88)( 91,103)( 92,104)(101,124)(102,123)(107,108);
s2 := Sym(128)!(  1, 42)(  2, 41)(  3, 43)(  4, 44)(  5, 46)(  6, 45)(  7, 47)(  8, 48)(  9, 34)( 10, 33)( 11, 35)( 12, 36)( 13, 38)( 14, 37)( 15, 39)( 16, 40)( 17, 58)( 18, 57)( 19, 59)( 20, 60)( 21, 62)( 22, 61)( 23, 63)( 24, 64)( 25, 50)( 26, 49)( 27, 51)( 28, 52)( 29, 54)( 30, 53)( 31, 55)( 32, 56)( 65,106)( 66,105)( 67,107)( 68,108)( 69,110)( 70,109)( 71,111)( 72,112)( 73, 98)( 74, 97)( 75, 99)( 76,100)( 77,102)( 78,101)( 79,103)( 80,104)( 81,122)( 82,121)( 83,123)( 84,124)( 85,126)( 86,125)( 87,127)( 88,128)( 89,114)( 90,113)( 91,115)( 92,116)( 93,118)( 94,117)( 95,119)( 96,120);
poly := sub<Sym(128)|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, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2 >;

References

None.

to this polytope.

Twisty Puzzle