Polytope of Type {3,8,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,8,6}*576
if this polytope has a name.
Group : SmallGroup(576,8340)
Rank : 4
Schlafli Type : {3,8,6}
Number of vertices, edges, etc : 6, 24, 48, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,8,6,2} of size 1152
   {3,8,6,3} of size 1728
Vertex Figure Of :
   {2,3,8,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,4,6}*288
   3-fold quotients : {3,8,2}*192
   6-fold quotients : {3,4,2}*96
   8-fold quotients : {3,2,6}*72
   12-fold quotients : {3,4,2}*48
   16-fold quotients : {3,2,3}*36
   24-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,8,12}*1152, {6,8,6}*1152b
   3-fold covers : {3,8,18}*1728, {9,8,6}*1728, {3,24,6}*1728a, {3,24,6}*1728b
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope