Polytope of Type {8,8} with group SmallGroup(256,722)

Overview

Group: SmallGroup(256,722)
Rank: 3
Schläfli Type: {8,8}
Vertices, edges, …: 16, 64, 16
Order of s₀s₁s₂: 8
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Dual
Self-Petrie

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

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

None.

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References