Polytope of Type {6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*576b
if this polytope has a name.
Group : SmallGroup(576,8328)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 48, 144, 48
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {6,6,2} of size 1152
Vertex Figure Of :
   {2,6,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6}*288b
   3-fold quotients : {6,6}*192b
   4-fold quotients : {3,6}*144
   6-fold quotients : {6,6}*96
   8-fold quotients : {6,6}*72c
   12-fold quotients : {3,6}*48, {6,3}*48
   16-fold quotients : {3,6}*36
   24-fold quotients : {3,3}*24, {6,2}*24
   48-fold quotients : {3,2}*12
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,6}*1152a, {6,12}*1152e
   3-fold covers : {18,6}*1728a, {6,6}*1728b, {6,6}*1728f
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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