Part of the Atlas of Small Regular Polytopes

Polytope of Type {8,4}

Atlas Canonical Name {8,4}*256d

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(256,6665)
Rank
3
Schläfli Type
{8,4}
Vertices, edges, …
32, 64, 16
Order of s0s1s2
8
Order of s0s1s2s1
8
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

16-fold

32-fold

Covers minimal covers in bold

2-fold

3-fold

5-fold

7-fold

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)^3*s1*s2> of order 2

12 facets

16 vertex figures

P/N, where N=<s0*s2*s1*s0*s1*s2> of order 4

8 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

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