Polytope of Type {2,56,6}

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

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