Part of the Atlas of Small Regular Polytopes

Polytope of Type {7,4}

Atlas Canonical Name {7,4}*1792a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1792,1083551)
Rank
3
Schläfli Type
{7,4}
Vertices, edges, …
224, 448, 128
Order of s0s1s2
7
Order of s0s1s2s1
8
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=<s0*s2*s1*s0*s1*(s2*s1*s0)^2*s2> of order 2

64 facets

120 vertex figures

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

64 facets

112 vertex figures

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

64 facets

112 vertex figures

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

32 facets

56 vertex figures

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

32 facets

64 vertex figures

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

32 facets

56 vertex figures

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

32 facets

60 vertex figures

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

32 facets

60 vertex figures

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

16 facets

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

References

None.

to this polytope.

Twisty Puzzle