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Polytope of Type {4,4,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,42}*1344
Also Known As : {{4,4|2},{4,42|2}}. if this polytope has another name.
Group : SmallGroup(1344,10968)
Rank : 4
Schlafli Type : {4,4,42}
Number of vertices, edges, etc : 4, 8, 84, 42
Order of s0s1s2s3 : 84
Order of s0s1s2s3s2s1 : 2
Special Properties :
   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 : {2,4,42}*672a, {4,2,42}*672
   3-fold quotients : {4,4,14}*448
   4-fold quotients : {4,2,21}*336, {2,2,42}*336
   6-fold quotients : {2,4,14}*224, {4,2,14}*224
   7-fold quotients : {4,4,6}*192
   8-fold quotients : {2,2,21}*168
   12-fold quotients : {4,2,7}*112, {2,2,14}*112
   14-fold quotients : {2,4,6}*96a, {4,2,6}*96
   21-fold quotients : {4,4,2}*64
   24-fold quotients : {2,2,7}*56
   28-fold quotients : {4,2,3}*48, {2,2,6}*48
   42-fold quotients : {2,4,2}*32, {4,2,2}*32
   56-fold quotients : {2,2,3}*24
   84-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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, 86)(  3, 87)(  4, 88)(  5, 89)(  6, 90)(  7, 91)(  8, 92)
(  9, 93)( 10, 94)( 11, 95)( 12, 96)( 13, 97)( 14, 98)( 15, 99)( 16,100)
( 17,101)( 18,102)( 19,103)( 20,104)( 21,105)( 22,106)( 23,107)( 24,108)
( 25,109)( 26,110)( 27,111)( 28,112)( 29,113)( 30,114)( 31,115)( 32,116)
( 33,117)( 34,118)( 35,119)( 36,120)( 37,121)( 38,122)( 39,123)( 40,124)
( 41,125)( 42,126)( 43,127)( 44,128)( 45,129)( 46,130)( 47,131)( 48,132)
( 49,133)( 50,134)( 51,135)( 52,136)( 53,137)( 54,138)( 55,139)( 56,140)
( 57,141)( 58,142)( 59,143)( 60,144)( 61,145)( 62,146)( 63,147)( 64,148)
( 65,149)( 66,150)( 67,151)( 68,152)( 69,153)( 70,154)( 71,155)( 72,156)
( 73,157)( 74,158)( 75,159)( 76,160)( 77,161)( 78,162)( 79,163)( 80,164)
( 81,165)( 82,166)( 83,167)( 84,168);;
s2 := (  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);;
s3 := (  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, 93)( 86, 92)( 87, 98)( 88, 97)( 89, 96)( 90, 95)( 91, 94)( 99,100)
(101,105)(102,104)(106,114)(107,113)(108,119)(109,118)(110,117)(111,116)
(112,115)(120,121)(122,126)(123,125)(127,135)(128,134)(129,140)(130,139)
(131,138)(132,137)(133,136)(141,142)(143,147)(144,146)(148,156)(149,155)
(150,161)(151,160)(152,159)(153,158)(154,157)(162,163)(164,168)(165,167);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, 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*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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, 86)(  3, 87)(  4, 88)(  5, 89)(  6, 90)(  7, 91)
(  8, 92)(  9, 93)( 10, 94)( 11, 95)( 12, 96)( 13, 97)( 14, 98)( 15, 99)
( 16,100)( 17,101)( 18,102)( 19,103)( 20,104)( 21,105)( 22,106)( 23,107)
( 24,108)( 25,109)( 26,110)( 27,111)( 28,112)( 29,113)( 30,114)( 31,115)
( 32,116)( 33,117)( 34,118)( 35,119)( 36,120)( 37,121)( 38,122)( 39,123)
( 40,124)( 41,125)( 42,126)( 43,127)( 44,128)( 45,129)( 46,130)( 47,131)
( 48,132)( 49,133)( 50,134)( 51,135)( 52,136)( 53,137)( 54,138)( 55,139)
( 56,140)( 57,141)( 58,142)( 59,143)( 60,144)( 61,145)( 62,146)( 63,147)
( 64,148)( 65,149)( 66,150)( 67,151)( 68,152)( 69,153)( 70,154)( 71,155)
( 72,156)( 73,157)( 74,158)( 75,159)( 76,160)( 77,161)( 78,162)( 79,163)
( 80,164)( 81,165)( 82,166)( 83,167)( 84,168);
s2 := 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);
s3 := 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, 93)( 86, 92)( 87, 98)( 88, 97)( 89, 96)( 90, 95)( 91, 94)
( 99,100)(101,105)(102,104)(106,114)(107,113)(108,119)(109,118)(110,117)
(111,116)(112,115)(120,121)(122,126)(123,125)(127,135)(128,134)(129,140)
(130,139)(131,138)(132,137)(133,136)(141,142)(143,147)(144,146)(148,156)
(149,155)(150,161)(151,160)(152,159)(153,158)(154,157)(162,163)(164,168)
(165,167);
poly := sub<Sym(168)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
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*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
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