Polytope of Type {9,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,4,4}*576a
if this polytope has a name.
Group : SmallGroup(576,1380)
Rank : 4
Schlafli Type : {9,4,4}
Number of vertices, edges, etc : 9, 36, 16, 8
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 4
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {9,4,4,2} of size 1152
   {9,4,4,3} of size 1728
Vertex Figure Of :
   {2,9,4,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,4,4}*192a
   4-fold quotients : {9,4,2}*144
   12-fold quotients : {3,4,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {9,4,4}*1152a, {9,4,4}*1152b, {18,4,4}*1152b, {18,4,4}*1152c
   3-fold covers : {27,4,4}*1728a
Permutation Representation (GAP) :

s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 41)( 22, 42)( 23, 44)( 24, 43)( 25, 37)( 26, 38)
( 27, 40)( 28, 39)( 29, 45)( 30, 46)( 31, 48)( 32, 47)( 49,113)( 50,114)
( 51,116)( 52,115)( 53,121)( 54,122)( 55,124)( 56,123)( 57,117)( 58,118)
( 59,120)( 60,119)( 61,125)( 62,126)( 63,128)( 64,127)( 65, 97)( 66, 98)
( 67,100)( 68, 99)( 69,105)( 70,106)( 71,108)( 72,107)( 73,101)( 74,102)
( 75,104)( 76,103)( 77,109)( 78,110)( 79,112)( 80,111)( 81,129)( 82,130)
( 83,132)( 84,131)( 85,137)( 86,138)( 87,140)( 88,139)( 89,133)( 90,134)
( 91,136)( 92,135)( 93,141)( 94,142)( 95,144)( 96,143);;
s1 := (  1, 49)(  2, 52)(  3, 51)(  4, 50)(  5, 53)(  6, 56)(  7, 55)(  8, 54)
(  9, 61)( 10, 64)( 11, 63)( 12, 62)( 13, 57)( 14, 60)( 15, 59)( 16, 58)
( 17, 81)( 18, 84)( 19, 83)( 20, 82)( 21, 85)( 22, 88)( 23, 87)( 24, 86)
( 25, 93)( 26, 96)( 27, 95)( 28, 94)( 29, 89)( 30, 92)( 31, 91)( 32, 90)
( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 37, 69)( 38, 72)( 39, 71)( 40, 70)
( 41, 77)( 42, 80)( 43, 79)( 44, 78)( 45, 73)( 46, 76)( 47, 75)( 48, 74)
( 97,113)( 98,116)( 99,115)(100,114)(101,117)(102,120)(103,119)(104,118)
(105,125)(106,128)(107,127)(108,126)(109,121)(110,124)(111,123)(112,122)
(130,132)(134,136)(137,141)(138,144)(139,143)(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);;
s3 := (  5,  7)(  6,  8)(  9, 12)( 10, 11)( 13, 14)( 15, 16)( 21, 23)( 22, 24)
( 25, 28)( 26, 27)( 29, 30)( 31, 32)( 37, 39)( 38, 40)( 41, 44)( 42, 43)
( 45, 46)( 47, 48)( 53, 55)( 54, 56)( 57, 60)( 58, 59)( 61, 62)( 63, 64)
( 69, 71)( 70, 72)( 73, 76)( 74, 75)( 77, 78)( 79, 80)( 85, 87)( 86, 88)
( 89, 92)( 90, 91)( 93, 94)( 95, 96)(101,103)(102,104)(105,108)(106,107)
(109,110)(111,112)(117,119)(118,120)(121,124)(122,123)(125,126)(127,128)
(133,135)(134,136)(137,140)(138,139)(141,142)(143,144);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s1*s0*s2*s1*s2*s1*s0*s1, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
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,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 41)( 22, 42)( 23, 44)( 24, 43)( 25, 37)
( 26, 38)( 27, 40)( 28, 39)( 29, 45)( 30, 46)( 31, 48)( 32, 47)( 49,113)
( 50,114)( 51,116)( 52,115)( 53,121)( 54,122)( 55,124)( 56,123)( 57,117)
( 58,118)( 59,120)( 60,119)( 61,125)( 62,126)( 63,128)( 64,127)( 65, 97)
( 66, 98)( 67,100)( 68, 99)( 69,105)( 70,106)( 71,108)( 72,107)( 73,101)
( 74,102)( 75,104)( 76,103)( 77,109)( 78,110)( 79,112)( 80,111)( 81,129)
( 82,130)( 83,132)( 84,131)( 85,137)( 86,138)( 87,140)( 88,139)( 89,133)
( 90,134)( 91,136)( 92,135)( 93,141)( 94,142)( 95,144)( 96,143);
s1 := Sym(144)!(  1, 49)(  2, 52)(  3, 51)(  4, 50)(  5, 53)(  6, 56)(  7, 55)
(  8, 54)(  9, 61)( 10, 64)( 11, 63)( 12, 62)( 13, 57)( 14, 60)( 15, 59)
( 16, 58)( 17, 81)( 18, 84)( 19, 83)( 20, 82)( 21, 85)( 22, 88)( 23, 87)
( 24, 86)( 25, 93)( 26, 96)( 27, 95)( 28, 94)( 29, 89)( 30, 92)( 31, 91)
( 32, 90)( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 37, 69)( 38, 72)( 39, 71)
( 40, 70)( 41, 77)( 42, 80)( 43, 79)( 44, 78)( 45, 73)( 46, 76)( 47, 75)
( 48, 74)( 97,113)( 98,116)( 99,115)(100,114)(101,117)(102,120)(103,119)
(104,118)(105,125)(106,128)(107,127)(108,126)(109,121)(110,124)(111,123)
(112,122)(130,132)(134,136)(137,141)(138,144)(139,143)(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);
s3 := Sym(144)!(  5,  7)(  6,  8)(  9, 12)( 10, 11)( 13, 14)( 15, 16)( 21, 23)
( 22, 24)( 25, 28)( 26, 27)( 29, 30)( 31, 32)( 37, 39)( 38, 40)( 41, 44)
( 42, 43)( 45, 46)( 47, 48)( 53, 55)( 54, 56)( 57, 60)( 58, 59)( 61, 62)
( 63, 64)( 69, 71)( 70, 72)( 73, 76)( 74, 75)( 77, 78)( 79, 80)( 85, 87)
( 86, 88)( 89, 92)( 90, 91)( 93, 94)( 95, 96)(101,103)(102,104)(105,108)
(106,107)(109,110)(111,112)(117,119)(118,120)(121,124)(122,123)(125,126)
(127,128)(133,135)(134,136)(137,140)(138,139)(141,142)(143,144);
poly := sub<Sym(144)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s1*s0*s2*s1*s2*s1*s0*s1, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope