Polytope of Type {6,6,8}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope