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Polytope of Type {56,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {56,6}*672
Also Known As : {56,6|2}. if this polytope has another name.
Group : SmallGroup(672,396)
Rank : 3
Schlafli Type : {56,6}
Number of vertices, edges, etc : 56, 168, 6
Order of s0s1s2 : 168
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {56,6,2} of size 1344
Vertex Figure Of :
   {2,56,6} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {28,6}*336a
   3-fold quotients : {56,2}*224
   4-fold quotients : {14,6}*168
   6-fold quotients : {28,2}*112
   7-fold quotients : {8,6}*96
   12-fold quotients : {14,2}*56
   14-fold quotients : {4,6}*48a
   21-fold quotients : {8,2}*32
   24-fold quotients : {7,2}*28
   28-fold quotients : {2,6}*24
   42-fold quotients : {4,2}*16
   56-fold quotients : {2,3}*12
   84-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {112,6}*1344, {56,12}*1344a
Permutation Representation (GAP) :
s0 := (  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)( 17, 20)
( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)( 37, 42)
( 38, 41)( 39, 40)( 43, 64)( 44, 70)( 45, 69)( 46, 68)( 47, 67)( 48, 66)
( 49, 65)( 50, 71)( 51, 77)( 52, 76)( 53, 75)( 54, 74)( 55, 73)( 56, 72)
( 57, 78)( 58, 84)( 59, 83)( 60, 82)( 61, 81)( 62, 80)( 63, 79)( 85,127)
( 86,133)( 87,132)( 88,131)( 89,130)( 90,129)( 91,128)( 92,134)( 93,140)
( 94,139)( 95,138)( 96,137)( 97,136)( 98,135)( 99,141)(100,147)(101,146)
(102,145)(103,144)(104,143)(105,142)(106,148)(107,154)(108,153)(109,152)
(110,151)(111,150)(112,149)(113,155)(114,161)(115,160)(116,159)(117,158)
(118,157)(119,156)(120,162)(121,168)(122,167)(123,166)(124,165)(125,164)
(126,163);;
s1 := (  1, 86)(  2, 85)(  3, 91)(  4, 90)(  5, 89)(  6, 88)(  7, 87)(  8,100)
(  9, 99)( 10,105)( 11,104)( 12,103)( 13,102)( 14,101)( 15, 93)( 16, 92)
( 17, 98)( 18, 97)( 19, 96)( 20, 95)( 21, 94)( 22,107)( 23,106)( 24,112)
( 25,111)( 26,110)( 27,109)( 28,108)( 29,121)( 30,120)( 31,126)( 32,125)
( 33,124)( 34,123)( 35,122)( 36,114)( 37,113)( 38,119)( 39,118)( 40,117)
( 41,116)( 42,115)( 43,149)( 44,148)( 45,154)( 46,153)( 47,152)( 48,151)
( 49,150)( 50,163)( 51,162)( 52,168)( 53,167)( 54,166)( 55,165)( 56,164)
( 57,156)( 58,155)( 59,161)( 60,160)( 61,159)( 62,158)( 63,157)( 64,128)
( 65,127)( 66,133)( 67,132)( 68,131)( 69,130)( 70,129)( 71,142)( 72,141)
( 73,147)( 74,146)( 75,145)( 76,144)( 77,143)( 78,135)( 79,134)( 80,140)
( 81,139)( 82,138)( 83,137)( 84,136);;
s2 := (  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)( 22, 29)
( 23, 30)( 24, 31)( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 43, 50)( 44, 51)
( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)( 66, 73)
( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85, 92)( 86, 93)( 87, 94)( 88, 95)
( 89, 96)( 90, 97)( 91, 98)(106,113)(107,114)(108,115)(109,116)(110,117)
(111,118)(112,119)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)
(133,140)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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(168)!(  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)
( 17, 20)( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)
( 37, 42)( 38, 41)( 39, 40)( 43, 64)( 44, 70)( 45, 69)( 46, 68)( 47, 67)
( 48, 66)( 49, 65)( 50, 71)( 51, 77)( 52, 76)( 53, 75)( 54, 74)( 55, 73)
( 56, 72)( 57, 78)( 58, 84)( 59, 83)( 60, 82)( 61, 81)( 62, 80)( 63, 79)
( 85,127)( 86,133)( 87,132)( 88,131)( 89,130)( 90,129)( 91,128)( 92,134)
( 93,140)( 94,139)( 95,138)( 96,137)( 97,136)( 98,135)( 99,141)(100,147)
(101,146)(102,145)(103,144)(104,143)(105,142)(106,148)(107,154)(108,153)
(109,152)(110,151)(111,150)(112,149)(113,155)(114,161)(115,160)(116,159)
(117,158)(118,157)(119,156)(120,162)(121,168)(122,167)(123,166)(124,165)
(125,164)(126,163);
s1 := Sym(168)!(  1, 86)(  2, 85)(  3, 91)(  4, 90)(  5, 89)(  6, 88)(  7, 87)
(  8,100)(  9, 99)( 10,105)( 11,104)( 12,103)( 13,102)( 14,101)( 15, 93)
( 16, 92)( 17, 98)( 18, 97)( 19, 96)( 20, 95)( 21, 94)( 22,107)( 23,106)
( 24,112)( 25,111)( 26,110)( 27,109)( 28,108)( 29,121)( 30,120)( 31,126)
( 32,125)( 33,124)( 34,123)( 35,122)( 36,114)( 37,113)( 38,119)( 39,118)
( 40,117)( 41,116)( 42,115)( 43,149)( 44,148)( 45,154)( 46,153)( 47,152)
( 48,151)( 49,150)( 50,163)( 51,162)( 52,168)( 53,167)( 54,166)( 55,165)
( 56,164)( 57,156)( 58,155)( 59,161)( 60,160)( 61,159)( 62,158)( 63,157)
( 64,128)( 65,127)( 66,133)( 67,132)( 68,131)( 69,130)( 70,129)( 71,142)
( 72,141)( 73,147)( 74,146)( 75,145)( 76,144)( 77,143)( 78,135)( 79,134)
( 80,140)( 81,139)( 82,138)( 83,137)( 84,136);
s2 := Sym(168)!(  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)
( 22, 29)( 23, 30)( 24, 31)( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 43, 50)
( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)
( 66, 73)( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85, 92)( 86, 93)( 87, 94)
( 88, 95)( 89, 96)( 90, 97)( 91, 98)(106,113)(107,114)(108,115)(109,116)
(110,117)(111,118)(112,119)(127,134)(128,135)(129,136)(130,137)(131,138)
(132,139)(133,140)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)
(154,161);
poly := sub<Sym(168)|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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
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