Polytope of Type {68,4} with group SmallGroup(1088,737)

Overview

Group: SmallGroup(1088,737)
Rank: 3
Schläfli Type: {68,4}
Vertices, edges, …: 136, 272, 8
Order of s₀s₁s₂: 68
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie

Quotients maximal quotients in bold

2-fold

{68,4}*544

4-fold

8-fold

{34,2}*136

16-fold

{17,2}*68

17-fold

{4,4}*64

34-fold

{4,4}*32

68-fold

136-fold

{2,2}*8

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=<(s1*s2)^2> of order 2

4 facets

4 of {68}*136

102 vertex figures

68 of {2}*4
34 of {4}*8

P/N, where N=<(s0*s1)^34> of order 2

6 facets

4 of {34}*68
2 of {68}*136

68 vertex figures

68 of {4}*8

Representations

Permutation Representation (GAP)

s0 := (  1, 69)(  2, 85)(  3, 84)(  4, 83)(  5, 82)(  6, 81)(  7, 80)(  8, 79)(  9, 78)( 10, 77)( 11, 76)( 12, 75)( 13, 74)( 14, 73)( 15, 72)( 16, 71)( 17, 70)( 18, 86)( 19,102)( 20,101)( 21,100)( 22, 99)( 23, 98)( 24, 97)( 25, 96)( 26, 95)( 27, 94)( 28, 93)( 29, 92)( 30, 91)( 31, 90)( 32, 89)( 33, 88)( 34, 87)( 35,103)( 36,119)( 37,118)( 38,117)( 39,116)( 40,115)( 41,114)( 42,113)( 43,112)( 44,111)( 45,110)( 46,109)( 47,108)( 48,107)( 49,106)( 50,105)( 51,104)( 52,120)( 53,136)( 54,135)( 55,134)( 56,133)( 57,132)( 58,131)( 59,130)( 60,129)( 61,128)( 62,127)( 63,126)( 64,125)( 65,124)( 66,123)( 67,122)( 68,121);;
s1 := (  1,  2)(  3, 17)(  4, 16)(  5, 15)(  6, 14)(  7, 13)(  8, 12)(  9, 11)( 18, 19)( 20, 34)( 21, 33)( 22, 32)( 23, 31)( 24, 30)( 25, 29)( 26, 28)( 35, 36)( 37, 51)( 38, 50)( 39, 49)( 40, 48)( 41, 47)( 42, 46)( 43, 45)( 52, 53)( 54, 68)( 55, 67)( 56, 66)( 57, 65)( 58, 64)( 59, 63)( 60, 62)( 69,104)( 70,103)( 71,119)( 72,118)( 73,117)( 74,116)( 75,115)( 76,114)( 77,113)( 78,112)( 79,111)( 80,110)( 81,109)( 82,108)( 83,107)( 84,106)( 85,105)( 86,121)( 87,120)( 88,136)( 89,135)( 90,134)( 91,133)( 92,132)( 93,131)( 94,130)( 95,129)( 96,128)( 97,127)( 98,126)( 99,125)(100,124)(101,123)(102,122);;
s2 := ( 35, 52)( 36, 53)( 37, 54)( 38, 55)( 39, 56)( 40, 57)( 41, 58)( 42, 59)( 43, 60)( 44, 61)( 45, 62)( 46, 63)( 47, 64)( 48, 65)( 49, 66)( 50, 67)( 51, 68)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136);;
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*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(136)!(  1, 69)(  2, 85)(  3, 84)(  4, 83)(  5, 82)(  6, 81)(  7, 80)(  8, 79)(  9, 78)( 10, 77)( 11, 76)( 12, 75)( 13, 74)( 14, 73)( 15, 72)( 16, 71)( 17, 70)( 18, 86)( 19,102)( 20,101)( 21,100)( 22, 99)( 23, 98)( 24, 97)( 25, 96)( 26, 95)( 27, 94)( 28, 93)( 29, 92)( 30, 91)( 31, 90)( 32, 89)( 33, 88)( 34, 87)( 35,103)( 36,119)( 37,118)( 38,117)( 39,116)( 40,115)( 41,114)( 42,113)( 43,112)( 44,111)( 45,110)( 46,109)( 47,108)( 48,107)( 49,106)( 50,105)( 51,104)( 52,120)( 53,136)( 54,135)( 55,134)( 56,133)( 57,132)( 58,131)( 59,130)( 60,129)( 61,128)( 62,127)( 63,126)( 64,125)( 65,124)( 66,123)( 67,122)( 68,121);
s1 := Sym(136)!(  1,  2)(  3, 17)(  4, 16)(  5, 15)(  6, 14)(  7, 13)(  8, 12)(  9, 11)( 18, 19)( 20, 34)( 21, 33)( 22, 32)( 23, 31)( 24, 30)( 25, 29)( 26, 28)( 35, 36)( 37, 51)( 38, 50)( 39, 49)( 40, 48)( 41, 47)( 42, 46)( 43, 45)( 52, 53)( 54, 68)( 55, 67)( 56, 66)( 57, 65)( 58, 64)( 59, 63)( 60, 62)( 69,104)( 70,103)( 71,119)( 72,118)( 73,117)( 74,116)( 75,115)( 76,114)( 77,113)( 78,112)( 79,111)( 80,110)( 81,109)( 82,108)( 83,107)( 84,106)( 85,105)( 86,121)( 87,120)( 88,136)( 89,135)( 90,134)( 91,133)( 92,132)( 93,131)( 94,130)( 95,129)( 96,128)( 97,127)( 98,126)( 99,125)(100,124)(101,123)(102,122);
s2 := Sym(136)!( 35, 52)( 36, 53)( 37, 54)( 38, 55)( 39, 56)( 40, 57)( 41, 58)( 42, 59)( 43, 60)( 44, 61)( 45, 62)( 46, 63)( 47, 64)( 48, 65)( 49, 66)( 50, 67)( 51, 68)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136);
poly := sub<Sym(136)|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*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References

None.

to this polytope.