Polytope of Type {9,8}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope