Polytope of Type {2,6,28}

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

Permutation Representation (Magma) :

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

to this polytope