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Polytope of Type {2,8,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,8,9}*576
if this polytope has a name.
Group : SmallGroup(576,4979)
Rank : 4
Schlafli Type : {2,8,9}
Number of vertices, edges, etc : 2, 16, 72, 18
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,8,9,2} of size 1152
Vertex Figure Of :
   {2,2,8,9} of size 1152
   {3,2,8,9} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,9}*288
   3-fold quotients : {2,8,3}*192
   4-fold quotients : {2,4,9}*144
   6-fold quotients : {2,4,3}*96
   8-fold quotients : {2,2,9}*72
   12-fold quotients : {2,4,3}*48
   24-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,8,9}*1152, {2,8,18}*1152b
   3-fold covers : {2,8,27}*1728, {2,24,9}*1728, {6,8,9}*1728
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  3, 77)(  4, 78)(  5, 76)(  6, 75)(  7, 81)(  8, 82)(  9, 80)( 10, 79)
( 11, 85)( 12, 86)( 13, 84)( 14, 83)( 15, 89)( 16, 90)( 17, 88)( 18, 87)
( 19, 93)( 20, 94)( 21, 92)( 22, 91)( 23, 97)( 24, 98)( 25, 96)( 26, 95)
( 27,101)( 28,102)( 29,100)( 30, 99)( 31,105)( 32,106)( 33,104)( 34,103)
( 35,109)( 36,110)( 37,108)( 38,107)( 39,113)( 40,114)( 41,112)( 42,111)
( 43,117)( 44,118)( 45,116)( 46,115)( 47,121)( 48,122)( 49,120)( 50,119)
( 51,125)( 52,126)( 53,124)( 54,123)( 55,129)( 56,130)( 57,128)( 58,127)
( 59,133)( 60,134)( 61,132)( 62,131)( 63,137)( 64,138)( 65,136)( 66,135)
( 67,141)( 68,142)( 69,140)( 70,139)( 71,145)( 72,146)( 73,144)( 74,143);;
s2 := (  5,  7)(  6,  8)(  9, 10)( 11, 19)( 12, 20)( 13, 23)( 14, 24)( 15, 21)
( 16, 22)( 17, 26)( 18, 25)( 27, 59)( 28, 60)( 29, 63)( 30, 64)( 31, 61)
( 32, 62)( 33, 66)( 34, 65)( 35, 51)( 36, 52)( 37, 55)( 38, 56)( 39, 53)
( 40, 54)( 41, 58)( 42, 57)( 43, 67)( 44, 68)( 45, 71)( 46, 72)( 47, 69)
( 48, 70)( 49, 74)( 50, 73)( 75, 76)( 77, 80)( 78, 79)( 83, 92)( 84, 91)
( 85, 96)( 86, 95)( 87, 94)( 88, 93)( 89, 97)( 90, 98)( 99,132)(100,131)
(101,136)(102,135)(103,134)(104,133)(105,137)(106,138)(107,124)(108,123)
(109,128)(110,127)(111,126)(112,125)(113,129)(114,130)(115,140)(116,139)
(117,144)(118,143)(119,142)(120,141)(121,145)(122,146);;
s3 := (  3, 27)(  4, 28)(  5, 30)(  6, 29)(  7, 33)(  8, 34)(  9, 31)( 10, 32)
( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 49)( 16, 50)( 17, 47)( 18, 48)
( 19, 35)( 20, 36)( 21, 38)( 22, 37)( 23, 41)( 24, 42)( 25, 39)( 26, 40)
( 51, 59)( 52, 60)( 53, 62)( 54, 61)( 55, 65)( 56, 66)( 57, 63)( 58, 64)
( 69, 70)( 71, 73)( 72, 74)( 75,100)( 76, 99)( 77,101)( 78,102)( 79,106)
( 80,105)( 81,104)( 82,103)( 83,116)( 84,115)( 85,117)( 86,118)( 87,122)
( 88,121)( 89,120)( 90,119)( 91,108)( 92,107)( 93,109)( 94,110)( 95,114)
( 96,113)( 97,112)( 98,111)(123,132)(124,131)(125,133)(126,134)(127,138)
(128,137)(129,136)(130,135)(139,140)(143,146)(144,145);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(146)!(1,2);
s1 := Sym(146)!(  3, 77)(  4, 78)(  5, 76)(  6, 75)(  7, 81)(  8, 82)(  9, 80)
( 10, 79)( 11, 85)( 12, 86)( 13, 84)( 14, 83)( 15, 89)( 16, 90)( 17, 88)
( 18, 87)( 19, 93)( 20, 94)( 21, 92)( 22, 91)( 23, 97)( 24, 98)( 25, 96)
( 26, 95)( 27,101)( 28,102)( 29,100)( 30, 99)( 31,105)( 32,106)( 33,104)
( 34,103)( 35,109)( 36,110)( 37,108)( 38,107)( 39,113)( 40,114)( 41,112)
( 42,111)( 43,117)( 44,118)( 45,116)( 46,115)( 47,121)( 48,122)( 49,120)
( 50,119)( 51,125)( 52,126)( 53,124)( 54,123)( 55,129)( 56,130)( 57,128)
( 58,127)( 59,133)( 60,134)( 61,132)( 62,131)( 63,137)( 64,138)( 65,136)
( 66,135)( 67,141)( 68,142)( 69,140)( 70,139)( 71,145)( 72,146)( 73,144)
( 74,143);
s2 := Sym(146)!(  5,  7)(  6,  8)(  9, 10)( 11, 19)( 12, 20)( 13, 23)( 14, 24)
( 15, 21)( 16, 22)( 17, 26)( 18, 25)( 27, 59)( 28, 60)( 29, 63)( 30, 64)
( 31, 61)( 32, 62)( 33, 66)( 34, 65)( 35, 51)( 36, 52)( 37, 55)( 38, 56)
( 39, 53)( 40, 54)( 41, 58)( 42, 57)( 43, 67)( 44, 68)( 45, 71)( 46, 72)
( 47, 69)( 48, 70)( 49, 74)( 50, 73)( 75, 76)( 77, 80)( 78, 79)( 83, 92)
( 84, 91)( 85, 96)( 86, 95)( 87, 94)( 88, 93)( 89, 97)( 90, 98)( 99,132)
(100,131)(101,136)(102,135)(103,134)(104,133)(105,137)(106,138)(107,124)
(108,123)(109,128)(110,127)(111,126)(112,125)(113,129)(114,130)(115,140)
(116,139)(117,144)(118,143)(119,142)(120,141)(121,145)(122,146);
s3 := Sym(146)!(  3, 27)(  4, 28)(  5, 30)(  6, 29)(  7, 33)(  8, 34)(  9, 31)
( 10, 32)( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 49)( 16, 50)( 17, 47)
( 18, 48)( 19, 35)( 20, 36)( 21, 38)( 22, 37)( 23, 41)( 24, 42)( 25, 39)
( 26, 40)( 51, 59)( 52, 60)( 53, 62)( 54, 61)( 55, 65)( 56, 66)( 57, 63)
( 58, 64)( 69, 70)( 71, 73)( 72, 74)( 75,100)( 76, 99)( 77,101)( 78,102)
( 79,106)( 80,105)( 81,104)( 82,103)( 83,116)( 84,115)( 85,117)( 86,118)
( 87,122)( 88,121)( 89,120)( 90,119)( 91,108)( 92,107)( 93,109)( 94,110)
( 95,114)( 96,113)( 97,112)( 98,111)(123,132)(124,131)(125,133)(126,134)
(127,138)(128,137)(129,136)(130,135)(139,140)(143,146)(144,145);
poly := sub<Sym(146)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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