Polytope of Type {8,4} with group SmallGroup(256,6665)

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

{4,4}*128

4-fold

{4,4}*64

8-fold

{4,4}*32

16-fold

32-fold

{2,2}*8

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*(s1*s0*s2)^3*s1*s2> of order 2

8 facets

8 of {8}*16

16 vertex figures

16 of {4}*8

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.