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Polytope of Type {3,6,16}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,16}*576
if this polytope has a name.
Group : SmallGroup(576,2907)
Rank : 4
Schlafli Type : {3,6,16}
Number of vertices, edges, etc : 3, 9, 48, 16
Order of s0s1s2s3 : 48
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,16,2} of size 1152
Vertex Figure Of :
   {2,3,6,16} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,8}*288
   3-fold quotients : {3,2,16}*192
   4-fold quotients : {3,6,4}*144
   6-fold quotients : {3,2,8}*96
   8-fold quotients : {3,6,2}*72
   12-fold quotients : {3,2,4}*48
   24-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,32}*1152, {6,6,16}*1152c
   3-fold covers : {9,6,16}*1728, {3,6,16}*1728a, {3,6,48}*1728b
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,  5)(  2,  4)(  3,  6)(  7,  8)( 10, 14)( 11, 13)( 12, 15)( 16, 17)
( 19, 23)( 20, 22)( 21, 24)( 25, 26)( 28, 32)( 29, 31)( 30, 33)( 34, 35)
( 37, 41)( 38, 40)( 39, 42)( 43, 44)( 46, 50)( 47, 49)( 48, 51)( 52, 53)
( 55, 59)( 56, 58)( 57, 60)( 61, 62)( 64, 68)( 65, 67)( 66, 69)( 70, 71)
( 73, 77)( 74, 76)( 75, 78)( 79, 80)( 82, 86)( 83, 85)( 84, 87)( 88, 89)
( 91, 95)( 92, 94)( 93, 96)( 97, 98)(100,104)(101,103)(102,105)(106,107)
(109,113)(110,112)(111,114)(115,116)(118,122)(119,121)(120,123)(124,125)
(127,131)(128,130)(129,132)(133,134)(136,140)(137,139)(138,141)(142,143);;
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)( 73,109)( 74,111)( 75,110)( 76,112)( 77,114)( 78,113)( 79,115)
( 80,117)( 81,116)( 82,118)( 83,120)( 84,119)( 85,121)( 86,123)( 87,122)
( 88,124)( 89,126)( 90,125)( 91,136)( 92,138)( 93,137)( 94,139)( 95,141)
( 96,140)( 97,142)( 98,144)( 99,143)(100,127)(101,129)(102,128)(103,130)
(104,132)(105,131)(106,133)(107,135)(108,134);;
s3 := (  1, 73)(  2, 74)(  3, 75)(  4, 76)(  5, 77)(  6, 78)(  7, 79)(  8, 80)
(  9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)( 16, 88)
( 17, 89)( 18, 90)( 19,100)( 20,101)( 21,102)( 22,103)( 23,104)( 24,105)
( 25,106)( 26,107)( 27,108)( 28, 91)( 29, 92)( 30, 93)( 31, 94)( 32, 95)
( 33, 96)( 34, 97)( 35, 98)( 36, 99)( 37,127)( 38,128)( 39,129)( 40,130)
( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)( 48,138)
( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,109)( 56,110)
( 57,111)( 58,112)( 59,113)( 60,114)( 61,115)( 62,116)( 63,117)( 64,118)
( 65,119)( 66,120)( 67,121)( 68,122)( 69,123)( 70,124)( 71,125)( 72,126);;
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*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*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(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,  5)(  2,  4)(  3,  6)(  7,  8)( 10, 14)( 11, 13)( 12, 15)
( 16, 17)( 19, 23)( 20, 22)( 21, 24)( 25, 26)( 28, 32)( 29, 31)( 30, 33)
( 34, 35)( 37, 41)( 38, 40)( 39, 42)( 43, 44)( 46, 50)( 47, 49)( 48, 51)
( 52, 53)( 55, 59)( 56, 58)( 57, 60)( 61, 62)( 64, 68)( 65, 67)( 66, 69)
( 70, 71)( 73, 77)( 74, 76)( 75, 78)( 79, 80)( 82, 86)( 83, 85)( 84, 87)
( 88, 89)( 91, 95)( 92, 94)( 93, 96)( 97, 98)(100,104)(101,103)(102,105)
(106,107)(109,113)(110,112)(111,114)(115,116)(118,122)(119,121)(120,123)
(124,125)(127,131)(128,130)(129,132)(133,134)(136,140)(137,139)(138,141)
(142,143);
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)( 73,109)( 74,111)( 75,110)( 76,112)( 77,114)( 78,113)
( 79,115)( 80,117)( 81,116)( 82,118)( 83,120)( 84,119)( 85,121)( 86,123)
( 87,122)( 88,124)( 89,126)( 90,125)( 91,136)( 92,138)( 93,137)( 94,139)
( 95,141)( 96,140)( 97,142)( 98,144)( 99,143)(100,127)(101,129)(102,128)
(103,130)(104,132)(105,131)(106,133)(107,135)(108,134);
s3 := Sym(144)!(  1, 73)(  2, 74)(  3, 75)(  4, 76)(  5, 77)(  6, 78)(  7, 79)
(  8, 80)(  9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)
( 16, 88)( 17, 89)( 18, 90)( 19,100)( 20,101)( 21,102)( 22,103)( 23,104)
( 24,105)( 25,106)( 26,107)( 27,108)( 28, 91)( 29, 92)( 30, 93)( 31, 94)
( 32, 95)( 33, 96)( 34, 97)( 35, 98)( 36, 99)( 37,127)( 38,128)( 39,129)
( 40,130)( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)
( 48,138)( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,109)
( 56,110)( 57,111)( 58,112)( 59,113)( 60,114)( 61,115)( 62,116)( 63,117)
( 64,118)( 65,119)( 66,120)( 67,121)( 68,122)( 69,123)( 70,124)( 71,125)
( 72,126);
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*s0*s1*s2*s1*s2*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
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