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Polytope of Type {4,12,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,14}*1344a
Also Known As : {{4,12|2},{12,14|2}}. if this polytope has another name.
Group : SmallGroup(1344,7764)
Rank : 4
Schlafli Type : {4,12,14}
Number of vertices, edges, etc : 4, 24, 84, 14
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 : {2,12,14}*672, {4,6,14}*672a
   3-fold quotients : {4,4,14}*448
   4-fold quotients : {2,6,14}*336
   6-fold quotients : {2,4,14}*224, {4,2,14}*224
   7-fold quotients : {4,12,2}*192a
   12-fold quotients : {4,2,7}*112, {2,2,14}*112
   14-fold quotients : {2,12,2}*96, {4,6,2}*96a
   21-fold quotients : {4,4,2}*64
   24-fold quotients : {2,2,7}*56
   28-fold quotients : {2,6,2}*48
   42-fold quotients : {2,4,2}*32, {4,2,2}*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 := ( 85,127)( 86,128)( 87,129)( 88,130)( 89,131)( 90,132)( 91,133)( 92,134)
( 93,135)( 94,136)( 95,137)( 96,138)( 97,139)( 98,140)( 99,141)(100,142)
(101,143)(102,144)(103,145)(104,146)(105,147)(106,148)(107,149)(108,150)
(109,151)(110,152)(111,153)(112,154)(113,155)(114,156)(115,157)(116,158)
(117,159)(118,160)(119,161)(120,162)(121,163)(122,164)(123,165)(124,166)
(125,167)(126,168);;
s1 := (  1, 85)(  2, 86)(  3, 87)(  4, 88)(  5, 89)(  6, 90)(  7, 91)(  8, 99)
(  9,100)( 10,101)( 11,102)( 12,103)( 13,104)( 14,105)( 15, 92)( 16, 93)
( 17, 94)( 18, 95)( 19, 96)( 20, 97)( 21, 98)( 22,106)( 23,107)( 24,108)
( 25,109)( 26,110)( 27,111)( 28,112)( 29,120)( 30,121)( 31,122)( 32,123)
( 33,124)( 34,125)( 35,126)( 36,113)( 37,114)( 38,115)( 39,116)( 40,117)
( 41,118)( 42,119)( 43,127)( 44,128)( 45,129)( 46,130)( 47,131)( 48,132)
( 49,133)( 50,141)( 51,142)( 52,143)( 53,144)( 54,145)( 55,146)( 56,147)
( 57,134)( 58,135)( 59,136)( 60,137)( 61,138)( 62,139)( 63,140)( 64,148)
( 65,149)( 66,150)( 67,151)( 68,152)( 69,153)( 70,154)( 71,162)( 72,163)
( 73,164)( 74,165)( 75,166)( 76,167)( 77,168)( 78,155)( 79,156)( 80,157)
( 81,158)( 82,159)( 83,160)( 84,161);;
s2 := (  1,  8)(  2, 14)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)( 16, 21)
( 17, 20)( 18, 19)( 22, 29)( 23, 35)( 24, 34)( 25, 33)( 26, 32)( 27, 31)
( 28, 30)( 37, 42)( 38, 41)( 39, 40)( 43, 50)( 44, 56)( 45, 55)( 46, 54)
( 47, 53)( 48, 52)( 49, 51)( 58, 63)( 59, 62)( 60, 61)( 64, 71)( 65, 77)
( 66, 76)( 67, 75)( 68, 74)( 69, 73)( 70, 72)( 79, 84)( 80, 83)( 81, 82)
( 85,113)( 86,119)( 87,118)( 88,117)( 89,116)( 90,115)( 91,114)( 92,106)
( 93,112)( 94,111)( 95,110)( 96,109)( 97,108)( 98,107)( 99,120)(100,126)
(101,125)(102,124)(103,123)(104,122)(105,121)(127,155)(128,161)(129,160)
(130,159)(131,158)(132,157)(133,156)(134,148)(135,154)(136,153)(137,152)
(138,151)(139,150)(140,149)(141,162)(142,168)(143,167)(144,166)(145,165)
(146,164)(147,163);;
s3 := (  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)( 17, 21)
( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 30)( 31, 35)( 32, 34)( 36, 37)
( 38, 42)( 39, 41)( 43, 44)( 45, 49)( 46, 48)( 50, 51)( 52, 56)( 53, 55)
( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)( 73, 77)
( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85, 86)( 87, 91)( 88, 90)( 92, 93)
( 94, 98)( 95, 97)( 99,100)(101,105)(102,104)(106,107)(108,112)(109,111)
(113,114)(115,119)(116,118)(120,121)(122,126)(123,125)(127,128)(129,133)
(130,132)(134,135)(136,140)(137,139)(141,142)(143,147)(144,146)(148,149)
(150,154)(151,153)(155,156)(157,161)(158,160)(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*s3*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, 
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)!( 85,127)( 86,128)( 87,129)( 88,130)( 89,131)( 90,132)( 91,133)
( 92,134)( 93,135)( 94,136)( 95,137)( 96,138)( 97,139)( 98,140)( 99,141)
(100,142)(101,143)(102,144)(103,145)(104,146)(105,147)(106,148)(107,149)
(108,150)(109,151)(110,152)(111,153)(112,154)(113,155)(114,156)(115,157)
(116,158)(117,159)(118,160)(119,161)(120,162)(121,163)(122,164)(123,165)
(124,166)(125,167)(126,168);
s1 := Sym(168)!(  1, 85)(  2, 86)(  3, 87)(  4, 88)(  5, 89)(  6, 90)(  7, 91)
(  8, 99)(  9,100)( 10,101)( 11,102)( 12,103)( 13,104)( 14,105)( 15, 92)
( 16, 93)( 17, 94)( 18, 95)( 19, 96)( 20, 97)( 21, 98)( 22,106)( 23,107)
( 24,108)( 25,109)( 26,110)( 27,111)( 28,112)( 29,120)( 30,121)( 31,122)
( 32,123)( 33,124)( 34,125)( 35,126)( 36,113)( 37,114)( 38,115)( 39,116)
( 40,117)( 41,118)( 42,119)( 43,127)( 44,128)( 45,129)( 46,130)( 47,131)
( 48,132)( 49,133)( 50,141)( 51,142)( 52,143)( 53,144)( 54,145)( 55,146)
( 56,147)( 57,134)( 58,135)( 59,136)( 60,137)( 61,138)( 62,139)( 63,140)
( 64,148)( 65,149)( 66,150)( 67,151)( 68,152)( 69,153)( 70,154)( 71,162)
( 72,163)( 73,164)( 74,165)( 75,166)( 76,167)( 77,168)( 78,155)( 79,156)
( 80,157)( 81,158)( 82,159)( 83,160)( 84,161);
s2 := Sym(168)!(  1,  8)(  2, 14)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)
( 16, 21)( 17, 20)( 18, 19)( 22, 29)( 23, 35)( 24, 34)( 25, 33)( 26, 32)
( 27, 31)( 28, 30)( 37, 42)( 38, 41)( 39, 40)( 43, 50)( 44, 56)( 45, 55)
( 46, 54)( 47, 53)( 48, 52)( 49, 51)( 58, 63)( 59, 62)( 60, 61)( 64, 71)
( 65, 77)( 66, 76)( 67, 75)( 68, 74)( 69, 73)( 70, 72)( 79, 84)( 80, 83)
( 81, 82)( 85,113)( 86,119)( 87,118)( 88,117)( 89,116)( 90,115)( 91,114)
( 92,106)( 93,112)( 94,111)( 95,110)( 96,109)( 97,108)( 98,107)( 99,120)
(100,126)(101,125)(102,124)(103,123)(104,122)(105,121)(127,155)(128,161)
(129,160)(130,159)(131,158)(132,157)(133,156)(134,148)(135,154)(136,153)
(137,152)(138,151)(139,150)(140,149)(141,162)(142,168)(143,167)(144,166)
(145,165)(146,164)(147,163);
s3 := Sym(168)!(  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)
( 17, 21)( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 30)( 31, 35)( 32, 34)
( 36, 37)( 38, 42)( 39, 41)( 43, 44)( 45, 49)( 46, 48)( 50, 51)( 52, 56)
( 53, 55)( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)
( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85, 86)( 87, 91)( 88, 90)
( 92, 93)( 94, 98)( 95, 97)( 99,100)(101,105)(102,104)(106,107)(108,112)
(109,111)(113,114)(115,119)(116,118)(120,121)(122,126)(123,125)(127,128)
(129,133)(130,132)(134,135)(136,140)(137,139)(141,142)(143,147)(144,146)
(148,149)(150,154)(151,153)(155,156)(157,161)(158,160)(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*s3*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, 
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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