Polytope of Type {6,8} with group SmallGroup(288,873)

Overview

Group: SmallGroup(288,873)
Rank: 3
Schläfli Type: {6,8}
Vertices, edges, …: 18, 72, 24
Order of s₀s₁s₂: 8
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Related Polytopes

Constructions

Facet of

Vertex figure of

Quotients maximal quotients in bold

2-fold

{6,4}*144

4-fold

{6,4}*72

9-fold

{2,8}*32

18-fold

{2,4}*16

36-fold

{2,2}*8

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-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*s1*s2> of order 3

16 facets

4 of {6}*12
12 of {2}*4

6 vertex figures

6 of {8}*16

P/N, where N=<(s0*s1*s2*s1)^2> of order 3

8 facets

8 of {6}*12

6 vertex figures

6 of {8}*16

Representations

Permutation Representation (GAP)

s0 := (  1, 55)(  2, 57)(  3, 56)(  4, 61)(  5, 63)(  6, 62)(  7, 58)(  8, 60)(  9, 59)( 10, 64)( 11, 66)( 12, 65)( 13, 70)( 14, 72)( 15, 71)( 16, 67)( 17, 69)( 18, 68)( 19, 46)( 20, 48)( 21, 47)( 22, 52)( 23, 54)( 24, 53)( 25, 49)( 26, 51)( 27, 50)( 28, 37)( 29, 39)( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 73,127)( 74,129)( 75,128)( 76,133)( 77,135)( 78,134)( 79,130)( 80,132)( 81,131)( 82,136)( 83,138)( 84,137)( 85,142)( 86,144)( 87,143)( 88,139)( 89,141)( 90,140)( 91,118)( 92,120)( 93,119)( 94,124)( 95,126)( 96,125)( 97,121)( 98,123)( 99,122)(100,109)(101,111)(102,110)(103,115)(104,117)(105,116)(106,112)(107,114)(108,113);;
s1 := (  1,  4)(  2,  5)(  3,  6)( 10, 13)( 11, 14)( 12, 15)( 19, 31)( 20, 32)( 21, 33)( 22, 28)( 23, 29)( 24, 30)( 25, 34)( 26, 35)( 27, 36)( 37, 49)( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)( 45, 54)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)( 66, 69)( 73, 94)( 74, 95)( 75, 96)( 76, 91)( 77, 92)( 78, 93)( 79, 97)( 80, 98)( 81, 99)( 82,103)( 83,104)( 84,105)( 85,100)( 86,101)( 87,102)( 88,106)( 89,107)( 90,108)(109,139)(110,140)(111,141)(112,136)(113,137)(114,138)(115,142)(116,143)(117,144)(118,130)(119,131)(120,132)(121,127)(122,128)(123,129)(124,133)(125,134)(126,135);;
s2 := (  1, 73)(  2, 76)(  3, 79)(  4, 74)(  5, 77)(  6, 80)(  7, 75)(  8, 78)(  9, 81)( 10, 82)( 11, 85)( 12, 88)( 13, 83)( 14, 86)( 15, 89)( 16, 84)( 17, 87)( 18, 90)( 19,100)( 20,103)( 21,106)( 22,101)( 23,104)( 24,107)( 25,102)( 26,105)( 27,108)( 28, 91)( 29, 94)( 30, 97)( 31, 92)( 32, 95)( 33, 98)( 34, 93)( 35, 96)( 36, 99)( 37,118)( 38,121)( 39,124)( 40,119)( 41,122)( 42,125)( 43,120)( 44,123)( 45,126)( 46,109)( 47,112)( 48,115)( 49,110)( 50,113)( 51,116)( 52,111)( 53,114)( 54,117)( 55,127)( 56,130)( 57,133)( 58,128)( 59,131)( 60,134)( 61,129)( 62,132)( 63,135)( 64,136)( 65,139)( 66,142)( 67,137)( 68,140)( 69,143)( 70,138)( 71,141)( 72,144);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*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(144)!(  1, 55)(  2, 57)(  3, 56)(  4, 61)(  5, 63)(  6, 62)(  7, 58)(  8, 60)(  9, 59)( 10, 64)( 11, 66)( 12, 65)( 13, 70)( 14, 72)( 15, 71)( 16, 67)( 17, 69)( 18, 68)( 19, 46)( 20, 48)( 21, 47)( 22, 52)( 23, 54)( 24, 53)( 25, 49)( 26, 51)( 27, 50)( 28, 37)( 29, 39)( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 73,127)( 74,129)( 75,128)( 76,133)( 77,135)( 78,134)( 79,130)( 80,132)( 81,131)( 82,136)( 83,138)( 84,137)( 85,142)( 86,144)( 87,143)( 88,139)( 89,141)( 90,140)( 91,118)( 92,120)( 93,119)( 94,124)( 95,126)( 96,125)( 97,121)( 98,123)( 99,122)(100,109)(101,111)(102,110)(103,115)(104,117)(105,116)(106,112)(107,114)(108,113);
s1 := Sym(144)!(  1,  4)(  2,  5)(  3,  6)( 10, 13)( 11, 14)( 12, 15)( 19, 31)( 20, 32)( 21, 33)( 22, 28)( 23, 29)( 24, 30)( 25, 34)( 26, 35)( 27, 36)( 37, 49)( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)( 45, 54)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)( 66, 69)( 73, 94)( 74, 95)( 75, 96)( 76, 91)( 77, 92)( 78, 93)( 79, 97)( 80, 98)( 81, 99)( 82,103)( 83,104)( 84,105)( 85,100)( 86,101)( 87,102)( 88,106)( 89,107)( 90,108)(109,139)(110,140)(111,141)(112,136)(113,137)(114,138)(115,142)(116,143)(117,144)(118,130)(119,131)(120,132)(121,127)(122,128)(123,129)(124,133)(125,134)(126,135);
s2 := Sym(144)!(  1, 73)(  2, 76)(  3, 79)(  4, 74)(  5, 77)(  6, 80)(  7, 75)(  8, 78)(  9, 81)( 10, 82)( 11, 85)( 12, 88)( 13, 83)( 14, 86)( 15, 89)( 16, 84)( 17, 87)( 18, 90)( 19,100)( 20,103)( 21,106)( 22,101)( 23,104)( 24,107)( 25,102)( 26,105)( 27,108)( 28, 91)( 29, 94)( 30, 97)( 31, 92)( 32, 95)( 33, 98)( 34, 93)( 35, 96)( 36, 99)( 37,118)( 38,121)( 39,124)( 40,119)( 41,122)( 42,125)( 43,120)( 44,123)( 45,126)( 46,109)( 47,112)( 48,115)( 49,110)( 50,113)( 51,116)( 52,111)( 53,114)( 54,117)( 55,127)( 56,130)( 57,133)( 58,128)( 59,131)( 60,134)( 61,129)( 62,132)( 63,135)( 64,136)( 65,139)( 66,142)( 67,137)( 68,140)( 69,143)( 70,138)( 71,141)( 72,144);
poly := sub<Sym(144)|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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References

None.

to this polytope.