Polytope of Type {4,9,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,9,4}*288
if this polytope has a name.
Group : SmallGroup(288,836)
Rank : 4
Schlafli Type : {4,9,4}
Number of vertices, edges, etc : 4, 18, 18, 4
Order of s₀s₁s₂s₃ : 9
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,9,4,2} of size 576
Vertex Figure Of :
   {2,4,9,4} of size 576
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,3,4}*96
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,9,4}*576a, {4,9,4}*576b, {4,18,4}*576d, {4,18,4}*576e, {4,18,4}*576f, {4,18,4}*576g
   3-fold covers : {4,27,4}*864
   4-fold covers : {4,36,4}*1152f, {4,36,4}*1152g, {4,36,4}*1152h, {4,36,4}*1152i, {4,9,8}*1152, {8,9,4}*1152, {4,9,4}*1152, {4,18,4}*1152c, {4,18,4}*1152d, {4,18,4}*1152e, {4,18,4}*1152f
   5-fold covers : {4,45,4}*1440
   6-fold covers : {4,27,4}*1728a, {4,27,4}*1728b, {4,54,4}*1728d, {4,54,4}*1728e, {4,54,4}*1728f, {4,54,4}*1728g, {4,9,12}*1728, {4,18,12}*1728e, {12,9,4}*1728, {12,18,4}*1728e
Permutation Representation (GAP) :

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

References : None.
to this polytope