Polytope of Type {16,8} with group SmallGroup(256,5312)

Overview

Group: SmallGroup(256,5312)
Rank: 3
Schläfli Type: {16,8}
Vertices, edges, …: 16, 64, 8
Order of s₀s₁s₂: 16
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat

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

References