Polytope of Type {2,84,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,84,4}*1344a
if this polytope has a name.
Group : SmallGroup(1344,10867)
Rank : 4
Schlafli Type : {2,84,4}
Number of vertices, edges, etc : 2, 84, 168, 4
Order of s₀s₁s₂s₃ : 84
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 : {2,84,2}*672, {2,42,4}*672a
   3-fold quotients : {2,28,4}*448
   4-fold quotients : {2,42,2}*336
   6-fold quotients : {2,28,2}*224, {2,14,4}*224
   7-fold quotients : {2,12,4}*192a
   8-fold quotients : {2,21,2}*168
   12-fold quotients : {2,14,2}*112
   14-fold quotients : {2,12,2}*96, {2,6,4}*96a
   21-fold quotients : {2,4,4}*64
   24-fold quotients : {2,7,2}*56
   28-fold quotients : {2,6,2}*48
   42-fold quotients : {2,2,4}*32, {2,4,2}*32
   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,  9)(  5,  8)(  6,  7)( 10, 17)( 11, 23)( 12, 22)( 13, 21)( 14, 20)
( 15, 19)( 16, 18)( 25, 30)( 26, 29)( 27, 28)( 31, 38)( 32, 44)( 33, 43)
( 34, 42)( 35, 41)( 36, 40)( 37, 39)( 46, 51)( 47, 50)( 48, 49)( 52, 59)
( 53, 65)( 54, 64)( 55, 63)( 56, 62)( 57, 61)( 58, 60)( 67, 72)( 68, 71)
( 69, 70)( 73, 80)( 74, 86)( 75, 85)( 76, 84)( 77, 83)( 78, 82)( 79, 81)
( 87,129)( 88,135)( 89,134)( 90,133)( 91,132)( 92,131)( 93,130)( 94,143)
( 95,149)( 96,148)( 97,147)( 98,146)( 99,145)(100,144)(101,136)(102,142)
(103,141)(104,140)(105,139)(106,138)(107,137)(108,150)(109,156)(110,155)
(111,154)(112,153)(113,152)(114,151)(115,164)(116,170)(117,169)(118,168)
(119,167)(120,166)(121,165)(122,157)(123,163)(124,162)(125,161)(126,160)
(127,159)(128,158);;
s2 := (  3, 95)(  4, 94)(  5,100)(  6, 99)(  7, 98)(  8, 97)(  9, 96)( 10, 88)
( 11, 87)( 12, 93)( 13, 92)( 14, 91)( 15, 90)( 16, 89)( 17,102)( 18,101)
( 19,107)( 20,106)( 21,105)( 22,104)( 23,103)( 24,116)( 25,115)( 26,121)
( 27,120)( 28,119)( 29,118)( 30,117)( 31,109)( 32,108)( 33,114)( 34,113)
( 35,112)( 36,111)( 37,110)( 38,123)( 39,122)( 40,128)( 41,127)( 42,126)
( 43,125)( 44,124)( 45,137)( 46,136)( 47,142)( 48,141)( 49,140)( 50,139)
( 51,138)( 52,130)( 53,129)( 54,135)( 55,134)( 56,133)( 57,132)( 58,131)
( 59,144)( 60,143)( 61,149)( 62,148)( 63,147)( 64,146)( 65,145)( 66,158)
( 67,157)( 68,163)( 69,162)( 70,161)( 71,160)( 72,159)( 73,151)( 74,150)
( 75,156)( 76,155)( 77,154)( 78,153)( 79,152)( 80,165)( 81,164)( 82,170)
( 83,169)( 84,168)( 85,167)( 86,166);;
s3 := ( 87,108)( 88,109)( 89,110)( 90,111)( 91,112)( 92,113)( 93,114)( 94,115)
( 95,116)( 96,117)( 97,118)( 98,119)( 99,120)(100,121)(101,122)(102,123)
(103,124)(104,125)(105,126)(106,127)(107,128)(129,150)(130,151)(131,152)
(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)
(140,161)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)
(148,169)(149,170);;
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*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, 
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*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)( 10, 17)( 11, 23)( 12, 22)( 13, 21)
( 14, 20)( 15, 19)( 16, 18)( 25, 30)( 26, 29)( 27, 28)( 31, 38)( 32, 44)
( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 37, 39)( 46, 51)( 47, 50)( 48, 49)
( 52, 59)( 53, 65)( 54, 64)( 55, 63)( 56, 62)( 57, 61)( 58, 60)( 67, 72)
( 68, 71)( 69, 70)( 73, 80)( 74, 86)( 75, 85)( 76, 84)( 77, 83)( 78, 82)
( 79, 81)( 87,129)( 88,135)( 89,134)( 90,133)( 91,132)( 92,131)( 93,130)
( 94,143)( 95,149)( 96,148)( 97,147)( 98,146)( 99,145)(100,144)(101,136)
(102,142)(103,141)(104,140)(105,139)(106,138)(107,137)(108,150)(109,156)
(110,155)(111,154)(112,153)(113,152)(114,151)(115,164)(116,170)(117,169)
(118,168)(119,167)(120,166)(121,165)(122,157)(123,163)(124,162)(125,161)
(126,160)(127,159)(128,158);
s2 := Sym(170)!(  3, 95)(  4, 94)(  5,100)(  6, 99)(  7, 98)(  8, 97)(  9, 96)
( 10, 88)( 11, 87)( 12, 93)( 13, 92)( 14, 91)( 15, 90)( 16, 89)( 17,102)
( 18,101)( 19,107)( 20,106)( 21,105)( 22,104)( 23,103)( 24,116)( 25,115)
( 26,121)( 27,120)( 28,119)( 29,118)( 30,117)( 31,109)( 32,108)( 33,114)
( 34,113)( 35,112)( 36,111)( 37,110)( 38,123)( 39,122)( 40,128)( 41,127)
( 42,126)( 43,125)( 44,124)( 45,137)( 46,136)( 47,142)( 48,141)( 49,140)
( 50,139)( 51,138)( 52,130)( 53,129)( 54,135)( 55,134)( 56,133)( 57,132)
( 58,131)( 59,144)( 60,143)( 61,149)( 62,148)( 63,147)( 64,146)( 65,145)
( 66,158)( 67,157)( 68,163)( 69,162)( 70,161)( 71,160)( 72,159)( 73,151)
( 74,150)( 75,156)( 76,155)( 77,154)( 78,153)( 79,152)( 80,165)( 81,164)
( 82,170)( 83,169)( 84,168)( 85,167)( 86,166);
s3 := Sym(170)!( 87,108)( 88,109)( 89,110)( 90,111)( 91,112)( 92,113)( 93,114)
( 94,115)( 95,116)( 96,117)( 97,118)( 98,119)( 99,120)(100,121)(101,122)
(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(129,150)(130,151)
(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)
(139,160)(140,161)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)
(147,168)(148,169)(149,170);
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*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, 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*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