Polytope of Type {6,56,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,56,2}*1344
if this polytope has a name.
Group : SmallGroup(1344,8483)
Rank : 4
Schlafli Type : {6,56,2}
Number of vertices, edges, etc : 6, 168, 56, 2
Order of s₀s₁s₂s₃ : 168
Order of s₀s₁s₂s₃s₂s₁ : 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 : {6,28,2}*672a
   3-fold quotients : {2,56,2}*448
   4-fold quotients : {6,14,2}*336
   6-fold quotients : {2,28,2}*224
   7-fold quotients : {6,8,2}*192
   12-fold quotients : {2,14,2}*112
   14-fold quotients : {6,4,2}*96a
   21-fold quotients : {2,8,2}*64
   24-fold quotients : {2,7,2}*56
   28-fold quotients : {6,2,2}*48
   42-fold quotients : {2,4,2}*32
   56-fold quotients : {3,2,2}*24
   84-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  8, 15)(  9, 16)( 10, 17)( 11, 18)( 12, 19)( 13, 20)( 14, 21)( 29, 36)
( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 50, 57)( 51, 58)
( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 71, 78)( 72, 79)( 73, 80)
( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 92, 99)( 93,100)( 94,101)( 95,102)
( 96,103)( 97,104)( 98,105)(113,120)(114,121)(115,122)(116,123)(117,124)
(118,125)(119,126)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)
(140,147)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168);;
s1 := (  1,  8)(  2, 14)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)( 16, 21)
( 17, 20)( 18, 19)( 22, 29)( 23, 35)( 24, 34)( 25, 33)( 26, 32)( 27, 31)
( 28, 30)( 37, 42)( 38, 41)( 39, 40)( 43, 71)( 44, 77)( 45, 76)( 46, 75)
( 47, 74)( 48, 73)( 49, 72)( 50, 64)( 51, 70)( 52, 69)( 53, 68)( 54, 67)
( 55, 66)( 56, 65)( 57, 78)( 58, 84)( 59, 83)( 60, 82)( 61, 81)( 62, 80)
( 63, 79)( 85,134)( 86,140)( 87,139)( 88,138)( 89,137)( 90,136)( 91,135)
( 92,127)( 93,133)( 94,132)( 95,131)( 96,130)( 97,129)( 98,128)( 99,141)
(100,147)(101,146)(102,145)(103,144)(104,143)(105,142)(106,155)(107,161)
(108,160)(109,159)(110,158)(111,157)(112,156)(113,148)(114,154)(115,153)
(116,152)(117,151)(118,150)(119,149)(120,162)(121,168)(122,167)(123,166)
(124,165)(125,164)(126,163);;
s2 := (  1, 86)(  2, 85)(  3, 91)(  4, 90)(  5, 89)(  6, 88)(  7, 87)(  8, 93)
(  9, 92)( 10, 98)( 11, 97)( 12, 96)( 13, 95)( 14, 94)( 15,100)( 16, 99)
( 17,105)( 18,104)( 19,103)( 20,102)( 21,101)( 22,107)( 23,106)( 24,112)
( 25,111)( 26,110)( 27,109)( 28,108)( 29,114)( 30,113)( 31,119)( 32,118)
( 33,117)( 34,116)( 35,115)( 36,121)( 37,120)( 38,126)( 39,125)( 40,124)
( 41,123)( 42,122)( 43,149)( 44,148)( 45,154)( 46,153)( 47,152)( 48,151)
( 49,150)( 50,156)( 51,155)( 52,161)( 53,160)( 54,159)( 55,158)( 56,157)
( 57,163)( 58,162)( 59,168)( 60,167)( 61,166)( 62,165)( 63,164)( 64,128)
( 65,127)( 66,133)( 67,132)( 68,131)( 69,130)( 70,129)( 71,135)( 72,134)
( 73,140)( 74,139)( 75,138)( 76,137)( 77,136)( 78,142)( 79,141)( 80,147)
( 81,146)( 82,145)( 83,144)( 84,143);;
s3 := (169,170);;
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, 
s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(170)!(  8, 15)(  9, 16)( 10, 17)( 11, 18)( 12, 19)( 13, 20)( 14, 21)
( 29, 36)( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 50, 57)
( 51, 58)( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 71, 78)( 72, 79)
( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 92, 99)( 93,100)( 94,101)
( 95,102)( 96,103)( 97,104)( 98,105)(113,120)(114,121)(115,122)(116,123)
(117,124)(118,125)(119,126)(134,141)(135,142)(136,143)(137,144)(138,145)
(139,146)(140,147)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)
(161,168);
s1 := Sym(170)!(  1,  8)(  2, 14)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)
( 16, 21)( 17, 20)( 18, 19)( 22, 29)( 23, 35)( 24, 34)( 25, 33)( 26, 32)
( 27, 31)( 28, 30)( 37, 42)( 38, 41)( 39, 40)( 43, 71)( 44, 77)( 45, 76)
( 46, 75)( 47, 74)( 48, 73)( 49, 72)( 50, 64)( 51, 70)( 52, 69)( 53, 68)
( 54, 67)( 55, 66)( 56, 65)( 57, 78)( 58, 84)( 59, 83)( 60, 82)( 61, 81)
( 62, 80)( 63, 79)( 85,134)( 86,140)( 87,139)( 88,138)( 89,137)( 90,136)
( 91,135)( 92,127)( 93,133)( 94,132)( 95,131)( 96,130)( 97,129)( 98,128)
( 99,141)(100,147)(101,146)(102,145)(103,144)(104,143)(105,142)(106,155)
(107,161)(108,160)(109,159)(110,158)(111,157)(112,156)(113,148)(114,154)
(115,153)(116,152)(117,151)(118,150)(119,149)(120,162)(121,168)(122,167)
(123,166)(124,165)(125,164)(126,163);
s2 := Sym(170)!(  1, 86)(  2, 85)(  3, 91)(  4, 90)(  5, 89)(  6, 88)(  7, 87)
(  8, 93)(  9, 92)( 10, 98)( 11, 97)( 12, 96)( 13, 95)( 14, 94)( 15,100)
( 16, 99)( 17,105)( 18,104)( 19,103)( 20,102)( 21,101)( 22,107)( 23,106)
( 24,112)( 25,111)( 26,110)( 27,109)( 28,108)( 29,114)( 30,113)( 31,119)
( 32,118)( 33,117)( 34,116)( 35,115)( 36,121)( 37,120)( 38,126)( 39,125)
( 40,124)( 41,123)( 42,122)( 43,149)( 44,148)( 45,154)( 46,153)( 47,152)
( 48,151)( 49,150)( 50,156)( 51,155)( 52,161)( 53,160)( 54,159)( 55,158)
( 56,157)( 57,163)( 58,162)( 59,168)( 60,167)( 61,166)( 62,165)( 63,164)
( 64,128)( 65,127)( 66,133)( 67,132)( 68,131)( 69,130)( 70,129)( 71,135)
( 72,134)( 73,140)( 74,139)( 75,138)( 76,137)( 77,136)( 78,142)( 79,141)
( 80,147)( 81,146)( 82,145)( 83,144)( 84,143);
s3 := Sym(170)!(169,170);
poly := sub<Sym(170)|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, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope