Polytope of Type {9,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,8}*576
if this polytope has a name.
Group : SmallGroup(576,1381)
Rank : 3
Schlafli Type : {9,8}
Number of vertices, edges, etc : 36, 144, 32
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {9,8,2} of size 1152
Vertex Figure Of :
   {2,9,8} of size 1152
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,8}*192
   4-fold quotients : {9,4}*144
   8-fold quotients : {9,4}*72
   12-fold quotients : {3,4}*48
   16-fold quotients : {9,2}*36
   24-fold quotients : {3,4}*24
   48-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {9,8}*1152, {18,8}*1152a, {18,8}*1152e
   3-fold covers : {27,8}*1728, {9,24}*1728
Permutation Representation (GAP) :

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

References : None.
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