Polytope of Type {4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4}*1152
Also Known As : {4,4}_(12,0), {4,4|12}. if this polytope has another name.
Group : SmallGroup(1152,32554)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 144, 288, 144
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4}*576
   4-fold quotients : {4,4}*288
   8-fold quotients : {4,4}*144
   9-fold quotients : {4,4}*128
   16-fold quotients : {4,4}*72
   18-fold quotients : {4,4}*64
   36-fold quotients : {4,4}*32
   72-fold quotients : {2,4}*16, {4,2}*16
   144-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope