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Polytope of Type {14,12,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,12,4}*1344a
Also Known As : {{14,12|2},{12,4|2}}. if this polytope has another name.
Group : SmallGroup(1344,7764)
Rank : 4
Schlafli Type : {14,12,4}
Number of vertices, edges, etc : 14, 84, 24, 4
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 : {14,12,2}*672, {14,6,4}*672a
   3-fold quotients : {14,4,4}*448
   4-fold quotients : {14,6,2}*336
   6-fold quotients : {14,2,4}*224, {14,4,2}*224
   7-fold quotients : {2,12,4}*192a
   12-fold quotients : {7,2,4}*112, {14,2,2}*112
   14-fold quotients : {2,12,2}*96, {2,6,4}*96a
   21-fold quotients : {2,4,4}*64
   24-fold quotients : {7,2,2}*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 := (  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)( 17, 20)
( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)( 37, 42)
( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)( 53, 54)
( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)( 73, 76)
( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)( 93, 98)
( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)(109,110)
(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)(129,132)
(130,131)(135,140)(136,139)(137,138)(142,147)(143,146)(144,145)(149,154)
(150,153)(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)(165,166);;
s1 := (  1,  2)(  3,  7)(  4,  6)(  8, 16)(  9, 15)( 10, 21)( 11, 20)( 12, 19)
( 13, 18)( 14, 17)( 22, 23)( 24, 28)( 25, 27)( 29, 37)( 30, 36)( 31, 42)
( 32, 41)( 33, 40)( 34, 39)( 35, 38)( 43, 44)( 45, 49)( 46, 48)( 50, 58)
( 51, 57)( 52, 63)( 53, 62)( 54, 61)( 55, 60)( 56, 59)( 64, 65)( 66, 70)
( 67, 69)( 71, 79)( 72, 78)( 73, 84)( 74, 83)( 75, 82)( 76, 81)( 77, 80)
( 85,107)( 86,106)( 87,112)( 88,111)( 89,110)( 90,109)( 91,108)( 92,121)
( 93,120)( 94,126)( 95,125)( 96,124)( 97,123)( 98,122)( 99,114)(100,113)
(101,119)(102,118)(103,117)(104,116)(105,115)(127,149)(128,148)(129,154)
(130,153)(131,152)(132,151)(133,150)(134,163)(135,162)(136,168)(137,167)
(138,166)(139,165)(140,164)(141,156)(142,155)(143,161)(144,160)(145,159)
(146,158)(147,157);;
s2 := (  1, 92)(  2, 93)(  3, 94)(  4, 95)(  5, 96)(  6, 97)(  7, 98)(  8, 85)
(  9, 86)( 10, 87)( 11, 88)( 12, 89)( 13, 90)( 14, 91)( 15, 99)( 16,100)
( 17,101)( 18,102)( 19,103)( 20,104)( 21,105)( 22,113)( 23,114)( 24,115)
( 25,116)( 26,117)( 27,118)( 28,119)( 29,106)( 30,107)( 31,108)( 32,109)
( 33,110)( 34,111)( 35,112)( 36,120)( 37,121)( 38,122)( 39,123)( 40,124)
( 41,125)( 42,126)( 43,134)( 44,135)( 45,136)( 46,137)( 47,138)( 48,139)
( 49,140)( 50,127)( 51,128)( 52,129)( 53,130)( 54,131)( 55,132)( 56,133)
( 57,141)( 58,142)( 59,143)( 60,144)( 61,145)( 62,146)( 63,147)( 64,155)
( 65,156)( 66,157)( 67,158)( 68,159)( 69,160)( 70,161)( 71,148)( 72,149)
( 73,150)( 74,151)( 75,152)( 76,153)( 77,154)( 78,162)( 79,163)( 80,164)
( 81,165)( 82,166)( 83,167)( 84,168);;
s3 := ( 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);;
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*s2*s1*s0*s1*s2*s1, 
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, 
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)(  9, 14)( 10, 13)( 11, 12)( 16, 21)
( 17, 20)( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)
( 37, 42)( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)
( 53, 54)( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)
( 73, 76)( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)
( 93, 98)( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)
(109,110)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)
(129,132)(130,131)(135,140)(136,139)(137,138)(142,147)(143,146)(144,145)
(149,154)(150,153)(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)
(165,166);
s1 := Sym(168)!(  1,  2)(  3,  7)(  4,  6)(  8, 16)(  9, 15)( 10, 21)( 11, 20)
( 12, 19)( 13, 18)( 14, 17)( 22, 23)( 24, 28)( 25, 27)( 29, 37)( 30, 36)
( 31, 42)( 32, 41)( 33, 40)( 34, 39)( 35, 38)( 43, 44)( 45, 49)( 46, 48)
( 50, 58)( 51, 57)( 52, 63)( 53, 62)( 54, 61)( 55, 60)( 56, 59)( 64, 65)
( 66, 70)( 67, 69)( 71, 79)( 72, 78)( 73, 84)( 74, 83)( 75, 82)( 76, 81)
( 77, 80)( 85,107)( 86,106)( 87,112)( 88,111)( 89,110)( 90,109)( 91,108)
( 92,121)( 93,120)( 94,126)( 95,125)( 96,124)( 97,123)( 98,122)( 99,114)
(100,113)(101,119)(102,118)(103,117)(104,116)(105,115)(127,149)(128,148)
(129,154)(130,153)(131,152)(132,151)(133,150)(134,163)(135,162)(136,168)
(137,167)(138,166)(139,165)(140,164)(141,156)(142,155)(143,161)(144,160)
(145,159)(146,158)(147,157);
s2 := Sym(168)!(  1, 92)(  2, 93)(  3, 94)(  4, 95)(  5, 96)(  6, 97)(  7, 98)
(  8, 85)(  9, 86)( 10, 87)( 11, 88)( 12, 89)( 13, 90)( 14, 91)( 15, 99)
( 16,100)( 17,101)( 18,102)( 19,103)( 20,104)( 21,105)( 22,113)( 23,114)
( 24,115)( 25,116)( 26,117)( 27,118)( 28,119)( 29,106)( 30,107)( 31,108)
( 32,109)( 33,110)( 34,111)( 35,112)( 36,120)( 37,121)( 38,122)( 39,123)
( 40,124)( 41,125)( 42,126)( 43,134)( 44,135)( 45,136)( 46,137)( 47,138)
( 48,139)( 49,140)( 50,127)( 51,128)( 52,129)( 53,130)( 54,131)( 55,132)
( 56,133)( 57,141)( 58,142)( 59,143)( 60,144)( 61,145)( 62,146)( 63,147)
( 64,155)( 65,156)( 66,157)( 67,158)( 68,159)( 69,160)( 70,161)( 71,148)
( 72,149)( 73,150)( 74,151)( 75,152)( 76,153)( 77,154)( 78,162)( 79,163)
( 80,164)( 81,165)( 82,166)( 83,167)( 84,168);
s3 := 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);
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*s2*s1*s0*s1*s2*s1, 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, 
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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