Polytope of Type {84,4}

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

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

Permutation Representation (Magma) :

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

References : None.
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