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Polytope of Type {3,2,8,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,8,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,46549)
Rank : 5
Schlafli Type : {3,2,8,6}
Number of vertices, edges, etc : 3, 3, 24, 72, 18
Order of s0s1s2s3s4 : 24
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,4,6}*864
   4-fold quotients : {3,2,4,6}*432
   9-fold quotients : {3,2,8,2}*192
   18-fold quotients : {3,2,4,2}*96
   36-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := (  7, 10)(  8, 11)(  9, 12)( 16, 19)( 17, 20)( 18, 21)( 22, 31)( 23, 32)
( 24, 33)( 25, 37)( 26, 38)( 27, 39)( 28, 34)( 29, 35)( 30, 36)( 40, 49)
( 41, 50)( 42, 51)( 43, 55)( 44, 56)( 45, 57)( 46, 52)( 47, 53)( 48, 54)
( 61, 64)( 62, 65)( 63, 66)( 70, 73)( 71, 74)( 72, 75)( 76, 94)( 77, 95)
( 78, 96)( 79,100)( 80,101)( 81,102)( 82, 97)( 83, 98)( 84, 99)( 85,103)
( 86,104)( 87,105)( 88,109)( 89,110)( 90,111)( 91,106)( 92,107)( 93,108)
(112,139)(113,140)(114,141)(115,145)(116,146)(117,147)(118,142)(119,143)
(120,144)(121,130)(122,131)(123,132)(124,136)(125,137)(126,138)(127,133)
(128,134)(129,135);;
s3 := (  4, 76)(  5, 79)(  6, 82)(  7, 77)(  8, 80)(  9, 83)( 10, 78)( 11, 81)
( 12, 84)( 13, 85)( 14, 88)( 15, 91)( 16, 86)( 17, 89)( 18, 92)( 19, 87)
( 20, 90)( 21, 93)( 22,103)( 23,106)( 24,109)( 25,104)( 26,107)( 27,110)
( 28,105)( 29,108)( 30,111)( 31, 94)( 32, 97)( 33,100)( 34, 95)( 35, 98)
( 36,101)( 37, 96)( 38, 99)( 39,102)( 40,121)( 41,124)( 42,127)( 43,122)
( 44,125)( 45,128)( 46,123)( 47,126)( 48,129)( 49,112)( 50,115)( 51,118)
( 52,113)( 53,116)( 54,119)( 55,114)( 56,117)( 57,120)( 58,130)( 59,133)
( 60,136)( 61,131)( 62,134)( 63,137)( 64,132)( 65,135)( 66,138)( 67,139)
( 68,142)( 69,145)( 70,140)( 71,143)( 72,146)( 73,141)( 74,144)( 75,147);;
s4 := (  4, 59)(  5, 58)(  6, 60)(  7, 65)(  8, 64)(  9, 66)( 10, 62)( 11, 61)
( 12, 63)( 13, 68)( 14, 67)( 15, 69)( 16, 74)( 17, 73)( 18, 75)( 19, 71)
( 20, 70)( 21, 72)( 22, 50)( 23, 49)( 24, 51)( 25, 56)( 26, 55)( 27, 57)
( 28, 53)( 29, 52)( 30, 54)( 31, 41)( 32, 40)( 33, 42)( 34, 47)( 35, 46)
( 36, 48)( 37, 44)( 38, 43)( 39, 45)( 76,131)( 77,130)( 78,132)( 79,137)
( 80,136)( 81,138)( 82,134)( 83,133)( 84,135)( 85,140)( 86,139)( 87,141)
( 88,146)( 89,145)( 90,147)( 91,143)( 92,142)( 93,144)( 94,122)( 95,121)
( 96,123)( 97,128)( 98,127)( 99,129)(100,125)(101,124)(102,126)(103,113)
(104,112)(105,114)(106,119)(107,118)(108,120)(109,116)(110,115)(111,117);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s4*s3*s4*s2*s3*s4*s2*s3*s4*s3*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(147)!(2,3);
s1 := Sym(147)!(1,2);
s2 := Sym(147)!(  7, 10)(  8, 11)(  9, 12)( 16, 19)( 17, 20)( 18, 21)( 22, 31)
( 23, 32)( 24, 33)( 25, 37)( 26, 38)( 27, 39)( 28, 34)( 29, 35)( 30, 36)
( 40, 49)( 41, 50)( 42, 51)( 43, 55)( 44, 56)( 45, 57)( 46, 52)( 47, 53)
( 48, 54)( 61, 64)( 62, 65)( 63, 66)( 70, 73)( 71, 74)( 72, 75)( 76, 94)
( 77, 95)( 78, 96)( 79,100)( 80,101)( 81,102)( 82, 97)( 83, 98)( 84, 99)
( 85,103)( 86,104)( 87,105)( 88,109)( 89,110)( 90,111)( 91,106)( 92,107)
( 93,108)(112,139)(113,140)(114,141)(115,145)(116,146)(117,147)(118,142)
(119,143)(120,144)(121,130)(122,131)(123,132)(124,136)(125,137)(126,138)
(127,133)(128,134)(129,135);
s3 := Sym(147)!(  4, 76)(  5, 79)(  6, 82)(  7, 77)(  8, 80)(  9, 83)( 10, 78)
( 11, 81)( 12, 84)( 13, 85)( 14, 88)( 15, 91)( 16, 86)( 17, 89)( 18, 92)
( 19, 87)( 20, 90)( 21, 93)( 22,103)( 23,106)( 24,109)( 25,104)( 26,107)
( 27,110)( 28,105)( 29,108)( 30,111)( 31, 94)( 32, 97)( 33,100)( 34, 95)
( 35, 98)( 36,101)( 37, 96)( 38, 99)( 39,102)( 40,121)( 41,124)( 42,127)
( 43,122)( 44,125)( 45,128)( 46,123)( 47,126)( 48,129)( 49,112)( 50,115)
( 51,118)( 52,113)( 53,116)( 54,119)( 55,114)( 56,117)( 57,120)( 58,130)
( 59,133)( 60,136)( 61,131)( 62,134)( 63,137)( 64,132)( 65,135)( 66,138)
( 67,139)( 68,142)( 69,145)( 70,140)( 71,143)( 72,146)( 73,141)( 74,144)
( 75,147);
s4 := Sym(147)!(  4, 59)(  5, 58)(  6, 60)(  7, 65)(  8, 64)(  9, 66)( 10, 62)
( 11, 61)( 12, 63)( 13, 68)( 14, 67)( 15, 69)( 16, 74)( 17, 73)( 18, 75)
( 19, 71)( 20, 70)( 21, 72)( 22, 50)( 23, 49)( 24, 51)( 25, 56)( 26, 55)
( 27, 57)( 28, 53)( 29, 52)( 30, 54)( 31, 41)( 32, 40)( 33, 42)( 34, 47)
( 35, 46)( 36, 48)( 37, 44)( 38, 43)( 39, 45)( 76,131)( 77,130)( 78,132)
( 79,137)( 80,136)( 81,138)( 82,134)( 83,133)( 84,135)( 85,140)( 86,139)
( 87,141)( 88,146)( 89,145)( 90,147)( 91,143)( 92,142)( 93,144)( 94,122)
( 95,121)( 96,123)( 97,128)( 98,127)( 99,129)(100,125)(101,124)(102,126)
(103,113)(104,112)(105,114)(106,119)(107,118)(108,120)(109,116)(110,115)
(111,117);
poly := sub<Sym(147)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s4*s3*s4*s2*s3*s4*s2*s3*s4*s3*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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