Polytope of Type {8,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,6}*576c
if this polytope has a name.
Group : SmallGroup(576,6980)
Rank : 4
Schlafli Type : {8,6,6}
Number of vertices, edges, etc : 8, 24, 18, 6
Order of s₀s₁s₂s₃ : 24
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,6,6,2} of size 1152
Vertex Figure Of :
   {2,8,6,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {8,6,3}*288, {4,6,6}*288c
   3-fold quotients : {8,2,6}*192
   4-fold quotients : {4,6,3}*144, {2,6,6}*144b
   6-fold quotients : {8,2,3}*96, {4,2,6}*96
   8-fold quotients : {2,6,3}*72
   9-fold quotients : {8,2,2}*64
   12-fold quotients : {4,2,3}*48, {2,2,6}*48
   18-fold quotients : {4,2,2}*32
   24-fold quotients : {2,2,3}*24
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,12,6}*1152c, {8,6,12}*1152c, {16,6,6}*1152c
   3-fold covers : {8,6,18}*1728b, {8,6,6}*1728c, {24,6,6}*1728e, {8,6,6}*1728e
Permutation Representation (GAP) :

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

References : None.
to this polytope