Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*576b
if this polytope has a name.
Group : SmallGroup(576,8659)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 8, 24, 24, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,6,2} of size 1152
   {6,6,6,3} of size 1728
Vertex Figure Of :
   {2,6,6,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,6}*288
   3-fold quotients : {6,6,2}*192
   6-fold quotients : {3,6,2}*96, {6,3,2}*96
   12-fold quotients : {3,3,2}*48, {2,2,6}*48
   24-fold quotients : {2,2,3}*24
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,6,12}*1152a, {6,12,6}*1152b, {12,6,6}*1152b, {6,12,6}*1152d, {12,6,6}*1152c, {6,6,6}*1152b
   3-fold covers : {6,6,18}*1728, {6,6,6}*1728d, {6,6,6}*1728f
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope