Polytope of Type {2,21,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,21,4}*672
if this polytope has a name.
Group : SmallGroup(672,1263)
Rank : 4
Schlafli Type : {2,21,4}
Number of vertices, edges, etc : 2, 42, 84, 8
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,21,4,2} of size 1344
Vertex Figure Of :
   {2,2,21,4} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,21,4}*336
   4-fold quotients : {2,21,2}*168
   7-fold quotients : {2,3,4}*96
   12-fold quotients : {2,7,2}*56
   14-fold quotients : {2,3,4}*48
   28-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,21,8}*1344, {2,42,4}*1344
Permutation Representation (GAP) :

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