Polytope of Type {3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6}*1764
Also Known As : {3,6}_(7,7). if this polytope has another name.
Group : SmallGroup(1764,146)
Rank : 3
Schlafli Type : {3,6}
Number of vertices, edges, etc : 147, 441, 294
Order of s₀s₁s₂ : 42
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,6}*588
   49-fold quotients : {3,6}*36
   147-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2, 43)(  3, 36)(  4, 29)(  5, 22)(  6, 15)(  7,  8)(  9, 49)( 10, 42)
( 11, 35)( 12, 28)( 13, 21)( 16, 48)( 17, 41)( 18, 34)( 19, 27)( 23, 47)
( 24, 40)( 25, 33)( 30, 46)( 31, 39)( 37, 45)( 50, 99)( 51,141)( 52,134)
( 53,127)( 54,120)( 55,113)( 56,106)( 57,105)( 58,147)( 59,140)( 60,133)
( 61,126)( 62,119)( 63,112)( 64,104)( 65,146)( 66,139)( 67,132)( 68,125)
( 69,118)( 70,111)( 71,103)( 72,145)( 73,138)( 74,131)( 75,124)( 76,117)
( 77,110)( 78,102)( 79,144)( 80,137)( 81,130)( 82,123)( 83,116)( 84,109)
( 85,101)( 86,143)( 87,136)( 88,129)( 89,122)( 90,115)( 91,108)( 92,100)
( 93,142)( 94,135)( 95,128)( 96,121)( 97,114)( 98,107);;
s1 := (  1, 50)(  2, 85)(  3, 71)(  4, 57)(  5, 92)(  6, 78)(  7, 64)(  8, 53)
(  9, 88)( 10, 74)( 11, 60)( 12, 95)( 13, 81)( 14, 67)( 15, 56)( 16, 91)
( 17, 77)( 18, 63)( 19, 98)( 20, 84)( 21, 70)( 22, 52)( 23, 87)( 24, 73)
( 25, 59)( 26, 94)( 27, 80)( 28, 66)( 29, 55)( 30, 90)( 31, 76)( 32, 62)
( 33, 97)( 34, 83)( 35, 69)( 36, 51)( 37, 86)( 38, 72)( 39, 58)( 40, 93)
( 41, 79)( 42, 65)( 43, 54)( 44, 89)( 45, 75)( 46, 61)( 47, 96)( 48, 82)
( 49, 68)(100,134)(101,120)(102,106)(103,141)(104,127)(105,113)(107,137)
(108,123)(110,144)(111,130)(112,116)(114,140)(115,126)(117,147)(118,133)
(121,136)(124,143)(125,129)(128,139)(131,146)(138,142);;
s2 := (  1, 44)(  3,  9)(  4, 16)(  5, 23)(  6, 30)(  7, 37)(  8, 45)( 11, 17)
( 12, 24)( 13, 31)( 14, 38)( 15, 46)( 19, 25)( 20, 32)( 21, 39)( 22, 47)
( 27, 33)( 28, 40)( 29, 48)( 35, 41)( 36, 49)( 50, 93)( 52, 58)( 53, 65)
( 54, 72)( 55, 79)( 56, 86)( 57, 94)( 60, 66)( 61, 73)( 62, 80)( 63, 87)
( 64, 95)( 68, 74)( 69, 81)( 70, 88)( 71, 96)( 76, 82)( 77, 89)( 78, 97)
( 84, 90)( 85, 98)( 99,142)(101,107)(102,114)(103,121)(104,128)(105,135)
(106,143)(109,115)(110,122)(111,129)(112,136)(113,144)(117,123)(118,130)
(119,137)(120,145)(125,131)(126,138)(127,146)(133,139)(134,147);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(147)!(  2, 43)(  3, 36)(  4, 29)(  5, 22)(  6, 15)(  7,  8)(  9, 49)
( 10, 42)( 11, 35)( 12, 28)( 13, 21)( 16, 48)( 17, 41)( 18, 34)( 19, 27)
( 23, 47)( 24, 40)( 25, 33)( 30, 46)( 31, 39)( 37, 45)( 50, 99)( 51,141)
( 52,134)( 53,127)( 54,120)( 55,113)( 56,106)( 57,105)( 58,147)( 59,140)
( 60,133)( 61,126)( 62,119)( 63,112)( 64,104)( 65,146)( 66,139)( 67,132)
( 68,125)( 69,118)( 70,111)( 71,103)( 72,145)( 73,138)( 74,131)( 75,124)
( 76,117)( 77,110)( 78,102)( 79,144)( 80,137)( 81,130)( 82,123)( 83,116)
( 84,109)( 85,101)( 86,143)( 87,136)( 88,129)( 89,122)( 90,115)( 91,108)
( 92,100)( 93,142)( 94,135)( 95,128)( 96,121)( 97,114)( 98,107);
s1 := Sym(147)!(  1, 50)(  2, 85)(  3, 71)(  4, 57)(  5, 92)(  6, 78)(  7, 64)
(  8, 53)(  9, 88)( 10, 74)( 11, 60)( 12, 95)( 13, 81)( 14, 67)( 15, 56)
( 16, 91)( 17, 77)( 18, 63)( 19, 98)( 20, 84)( 21, 70)( 22, 52)( 23, 87)
( 24, 73)( 25, 59)( 26, 94)( 27, 80)( 28, 66)( 29, 55)( 30, 90)( 31, 76)
( 32, 62)( 33, 97)( 34, 83)( 35, 69)( 36, 51)( 37, 86)( 38, 72)( 39, 58)
( 40, 93)( 41, 79)( 42, 65)( 43, 54)( 44, 89)( 45, 75)( 46, 61)( 47, 96)
( 48, 82)( 49, 68)(100,134)(101,120)(102,106)(103,141)(104,127)(105,113)
(107,137)(108,123)(110,144)(111,130)(112,116)(114,140)(115,126)(117,147)
(118,133)(121,136)(124,143)(125,129)(128,139)(131,146)(138,142);
s2 := Sym(147)!(  1, 44)(  3,  9)(  4, 16)(  5, 23)(  6, 30)(  7, 37)(  8, 45)
( 11, 17)( 12, 24)( 13, 31)( 14, 38)( 15, 46)( 19, 25)( 20, 32)( 21, 39)
( 22, 47)( 27, 33)( 28, 40)( 29, 48)( 35, 41)( 36, 49)( 50, 93)( 52, 58)
( 53, 65)( 54, 72)( 55, 79)( 56, 86)( 57, 94)( 60, 66)( 61, 73)( 62, 80)
( 63, 87)( 64, 95)( 68, 74)( 69, 81)( 70, 88)( 71, 96)( 76, 82)( 77, 89)
( 78, 97)( 84, 90)( 85, 98)( 99,142)(101,107)(102,114)(103,121)(104,128)
(105,135)(106,143)(109,115)(110,122)(111,129)(112,136)(113,144)(117,123)
(118,130)(119,137)(120,145)(125,131)(126,138)(127,146)(133,139)(134,147);
poly := sub<Sym(147)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;

References : None.
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