Polytope of Type {4,84}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,84}*672a
Also Known As : {4,84|2}. if this polytope has another name.
Group : SmallGroup(672,947)
Rank : 3
Schlafli Type : {4,84}
Number of vertices, edges, etc : 4, 168, 84
Order of s₀s₁s₂ : 84
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,84,2} of size 1344
Vertex Figure Of :
   {2,4,84} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,84}*336, {4,42}*336a
   3-fold quotients : {4,28}*224
   4-fold quotients : {2,42}*168
   6-fold quotients : {2,28}*112, {4,14}*112
   7-fold quotients : {4,12}*96a
   8-fold quotients : {2,21}*84
   12-fold quotients : {2,14}*56
   14-fold quotients : {2,12}*48, {4,6}*48a
   21-fold quotients : {4,4}*32
   24-fold quotients : {2,7}*28
   28-fold quotients : {2,6}*24
   42-fold quotients : {2,4}*16, {4,2}*16
   56-fold quotients : {2,3}*12
   84-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,168}*1344a, {4,84}*1344a, {4,168}*1344b, {8,84}*1344a, {8,84}*1344b
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, 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 >;

References : None.
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