Polytope of Type {8,3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,3,6}*576
if this polytope has a name.
Group : SmallGroup(576,8340)
Rank : 4
Schlafli Type : {8,3,6}
Number of vertices, edges, etc : 16, 24, 18, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,3,6,2} of size 1152
   {8,3,6,3} of size 1728
Vertex Figure Of :
   {2,8,3,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,3,6}*288
   3-fold quotients : {8,3,2}*192
   4-fold quotients : {4,3,6}*144
   6-fold quotients : {4,3,2}*96
   8-fold quotients : {2,3,6}*72
   12-fold quotients : {4,3,2}*48
   24-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,6,6}*1152c
   3-fold covers : {8,9,6}*1728, {8,3,6}*1728, {24,3,6}*1728
Permutation Representation (GAP) :

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

References : None.
to this polytope