Polytope of Type {8,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,9}*288
if this polytope has a name.
Group : SmallGroup(288,340)
Rank : 3
Schlafli Type : {8,9}
Number of vertices, edges, etc : 16, 72, 18
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,9,2} of size 576
   {8,9,4} of size 1152
   {8,9,6} of size 1728
Vertex Figure Of :
   {2,8,9} of size 576
   {4,8,9} of size 1152
   {6,8,9} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,9}*144
   3-fold quotients : {8,3}*96
   4-fold quotients : {4,9}*72
   6-fold quotients : {4,3}*48
   8-fold quotients : {2,9}*36
   12-fold quotients : {4,3}*24
   24-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,18}*576b
   3-fold covers : {8,27}*864, {24,9}*864
   4-fold covers : {8,9}*1152, {8,36}*1152e, {8,18}*1152f, {8,36}*1152h
   5-fold covers : {8,45}*1440
   6-fold covers : {8,54}*1728b, {24,18}*1728b, {24,18}*1728e
Permutation Representation (GAP) :

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

References : None.
to this polytope