Polytope of Type {2,28,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,28,6}*1344
if this polytope has a name.
Group : SmallGroup(1344,11695)
Rank : 4
Schlafli Type : {2,28,6}
Number of vertices, edges, etc : 2, 56, 168, 12
Order of s₀s₁s₂s₃ : 42
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}*672b
   4-fold quotients : {2,14,6}*336
   7-fold quotients : {2,4,6}*192
   12-fold quotients : {2,14,2}*112
   14-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
   24-fold quotients : {2,7,2}*56
   28-fold quotients : {2,4,3}*48, {2,2,6}*48
   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 := (  3,  5)(  4,  6)(  7, 29)(  8, 30)(  9, 27)( 10, 28)( 11, 25)( 12, 26)
( 13, 23)( 14, 24)( 15, 21)( 16, 22)( 17, 19)( 18, 20)( 31, 33)( 32, 34)
( 35, 57)( 36, 58)( 37, 55)( 38, 56)( 39, 53)( 40, 54)( 41, 51)( 42, 52)
( 43, 49)( 44, 50)( 45, 47)( 46, 48)( 59, 61)( 60, 62)( 63, 85)( 64, 86)
( 65, 83)( 66, 84)( 67, 81)( 68, 82)( 69, 79)( 70, 80)( 71, 77)( 72, 78)
( 73, 75)( 74, 76)( 87, 89)( 88, 90)( 91,113)( 92,114)( 93,111)( 94,112)
( 95,109)( 96,110)( 97,107)( 98,108)( 99,105)(100,106)(101,103)(102,104)
(115,117)(116,118)(119,141)(120,142)(121,139)(122,140)(123,137)(124,138)
(125,135)(126,136)(127,133)(128,134)(129,131)(130,132)(143,145)(144,146)
(147,169)(148,170)(149,167)(150,168)(151,165)(152,166)(153,163)(154,164)
(155,161)(156,162)(157,159)(158,160);;
s2 := (  3,  7)(  4,  9)(  5,  8)(  6, 10)( 11, 27)( 12, 29)( 13, 28)( 14, 30)
( 15, 23)( 16, 25)( 17, 24)( 18, 26)( 20, 21)( 31, 63)( 32, 65)( 33, 64)
( 34, 66)( 35, 59)( 36, 61)( 37, 60)( 38, 62)( 39, 83)( 40, 85)( 41, 84)
( 42, 86)( 43, 79)( 44, 81)( 45, 80)( 46, 82)( 47, 75)( 48, 77)( 49, 76)
( 50, 78)( 51, 71)( 52, 73)( 53, 72)( 54, 74)( 55, 67)( 56, 69)( 57, 68)
( 58, 70)( 87, 91)( 88, 93)( 89, 92)( 90, 94)( 95,111)( 96,113)( 97,112)
( 98,114)( 99,107)(100,109)(101,108)(102,110)(104,105)(115,147)(116,149)
(117,148)(118,150)(119,143)(120,145)(121,144)(122,146)(123,167)(124,169)
(125,168)(126,170)(127,163)(128,165)(129,164)(130,166)(131,159)(132,161)
(133,160)(134,162)(135,155)(136,157)(137,156)(138,158)(139,151)(140,153)
(141,152)(142,154);;
s3 := (  3,143)(  4,146)(  5,145)(  6,144)(  7,147)(  8,150)(  9,149)( 10,148)
( 11,151)( 12,154)( 13,153)( 14,152)( 15,155)( 16,158)( 17,157)( 18,156)
( 19,159)( 20,162)( 21,161)( 22,160)( 23,163)( 24,166)( 25,165)( 26,164)
( 27,167)( 28,170)( 29,169)( 30,168)( 31,115)( 32,118)( 33,117)( 34,116)
( 35,119)( 36,122)( 37,121)( 38,120)( 39,123)( 40,126)( 41,125)( 42,124)
( 43,127)( 44,130)( 45,129)( 46,128)( 47,131)( 48,134)( 49,133)( 50,132)
( 51,135)( 52,138)( 53,137)( 54,136)( 55,139)( 56,142)( 57,141)( 58,140)
( 59, 87)( 60, 90)( 61, 89)( 62, 88)( 63, 91)( 64, 94)( 65, 93)( 66, 92)
( 67, 95)( 68, 98)( 69, 97)( 70, 96)( 71, 99)( 72,102)( 73,101)( 74,100)
( 75,103)( 76,106)( 77,105)( 78,104)( 79,107)( 80,110)( 81,109)( 82,108)
( 83,111)( 84,114)( 85,113)( 86,112);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2, 
s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope