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Polytope of Type {8,6,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,2}*576
if this polytope has a name.
Group : SmallGroup(576,8380)
Rank : 4
Schlafli Type : {8,6,2}
Number of vertices, edges, etc : 24, 72, 18, 2
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,6,2,2} of size 1152
   {8,6,2,3} of size 1728
Vertex Figure Of :
   {2,8,6,2} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6,2}*288
   4-fold quotients : {4,6,2}*144
   9-fold quotients : {8,2,2}*64
   18-fold quotients : {4,2,2}*32
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,12,2}*1152a, {8,6,4}*1152b, {16,6,2}*1152
   3-fold covers : {8,6,2}*1728a, {24,6,2}*1728d, {24,6,2}*1728e, {8,6,6}*1728f, {8,6,2}*1728b, {24,6,2}*1728g, {24,6,2}*1728h
Permutation Representation (GAP) :
s0 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 19, 28)( 20, 29)
( 21, 30)( 22, 34)( 23, 35)( 24, 36)( 25, 31)( 26, 32)( 27, 33)( 37, 46)
( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)
( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 73, 91)( 74, 92)
( 75, 93)( 76, 97)( 77, 98)( 78, 99)( 79, 94)( 80, 95)( 81, 96)( 82,100)
( 83,101)( 84,102)( 85,106)( 86,107)( 87,108)( 88,103)( 89,104)( 90,105)
(109,136)(110,137)(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)
(117,141)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,130)
(125,131)(126,132);;
s1 := (  1, 73)(  2, 76)(  3, 79)(  4, 74)(  5, 77)(  6, 80)(  7, 75)(  8, 78)
(  9, 81)( 10, 82)( 11, 85)( 12, 88)( 13, 83)( 14, 86)( 15, 89)( 16, 84)
( 17, 87)( 18, 90)( 19,100)( 20,103)( 21,106)( 22,101)( 23,104)( 24,107)
( 25,102)( 26,105)( 27,108)( 28, 91)( 29, 94)( 30, 97)( 31, 92)( 32, 95)
( 33, 98)( 34, 93)( 35, 96)( 36, 99)( 37,118)( 38,121)( 39,124)( 40,119)
( 41,122)( 42,125)( 43,120)( 44,123)( 45,126)( 46,109)( 47,112)( 48,115)
( 49,110)( 50,113)( 51,116)( 52,111)( 53,114)( 54,117)( 55,127)( 56,130)
( 57,133)( 58,128)( 59,131)( 60,134)( 61,129)( 62,132)( 63,135)( 64,136)
( 65,139)( 66,142)( 67,137)( 68,140)( 69,143)( 70,138)( 71,141)( 72,144);;
s2 := (  1, 56)(  2, 55)(  3, 57)(  4, 62)(  5, 61)(  6, 63)(  7, 59)(  8, 58)
(  9, 60)( 10, 65)( 11, 64)( 12, 66)( 13, 71)( 14, 70)( 15, 72)( 16, 68)
( 17, 67)( 18, 69)( 19, 47)( 20, 46)( 21, 48)( 22, 53)( 23, 52)( 24, 54)
( 25, 50)( 26, 49)( 27, 51)( 28, 38)( 29, 37)( 30, 39)( 31, 44)( 32, 43)
( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 73,128)( 74,127)( 75,129)( 76,134)
( 77,133)( 78,135)( 79,131)( 80,130)( 81,132)( 82,137)( 83,136)( 84,138)
( 85,143)( 86,142)( 87,144)( 88,140)( 89,139)( 90,141)( 91,119)( 92,118)
( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)( 98,121)( 99,123)(100,110)
(101,109)(102,111)(103,116)(104,115)(105,117)(106,113)(107,112)(108,114);;
s3 := (145,146);;
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, s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*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(146)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 19, 28)
( 20, 29)( 21, 30)( 22, 34)( 23, 35)( 24, 36)( 25, 31)( 26, 32)( 27, 33)
( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)
( 45, 51)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 73, 91)
( 74, 92)( 75, 93)( 76, 97)( 77, 98)( 78, 99)( 79, 94)( 80, 95)( 81, 96)
( 82,100)( 83,101)( 84,102)( 85,106)( 86,107)( 87,108)( 88,103)( 89,104)
( 90,105)(109,136)(110,137)(111,138)(112,142)(113,143)(114,144)(115,139)
(116,140)(117,141)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)
(124,130)(125,131)(126,132);
s1 := Sym(146)!(  1, 73)(  2, 76)(  3, 79)(  4, 74)(  5, 77)(  6, 80)(  7, 75)
(  8, 78)(  9, 81)( 10, 82)( 11, 85)( 12, 88)( 13, 83)( 14, 86)( 15, 89)
( 16, 84)( 17, 87)( 18, 90)( 19,100)( 20,103)( 21,106)( 22,101)( 23,104)
( 24,107)( 25,102)( 26,105)( 27,108)( 28, 91)( 29, 94)( 30, 97)( 31, 92)
( 32, 95)( 33, 98)( 34, 93)( 35, 96)( 36, 99)( 37,118)( 38,121)( 39,124)
( 40,119)( 41,122)( 42,125)( 43,120)( 44,123)( 45,126)( 46,109)( 47,112)
( 48,115)( 49,110)( 50,113)( 51,116)( 52,111)( 53,114)( 54,117)( 55,127)
( 56,130)( 57,133)( 58,128)( 59,131)( 60,134)( 61,129)( 62,132)( 63,135)
( 64,136)( 65,139)( 66,142)( 67,137)( 68,140)( 69,143)( 70,138)( 71,141)
( 72,144);
s2 := Sym(146)!(  1, 56)(  2, 55)(  3, 57)(  4, 62)(  5, 61)(  6, 63)(  7, 59)
(  8, 58)(  9, 60)( 10, 65)( 11, 64)( 12, 66)( 13, 71)( 14, 70)( 15, 72)
( 16, 68)( 17, 67)( 18, 69)( 19, 47)( 20, 46)( 21, 48)( 22, 53)( 23, 52)
( 24, 54)( 25, 50)( 26, 49)( 27, 51)( 28, 38)( 29, 37)( 30, 39)( 31, 44)
( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 73,128)( 74,127)( 75,129)
( 76,134)( 77,133)( 78,135)( 79,131)( 80,130)( 81,132)( 82,137)( 83,136)
( 84,138)( 85,143)( 86,142)( 87,144)( 88,140)( 89,139)( 90,141)( 91,119)
( 92,118)( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)( 98,121)( 99,123)
(100,110)(101,109)(102,111)(103,116)(104,115)(105,117)(106,113)(107,112)
(108,114);
s3 := Sym(146)!(145,146);
poly := sub<Sym(146)|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, 
s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 

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