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Polytope of Type {12,4,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4,14}*1344
Also Known As : {{12,4|2},{4,14|2}}. if this polytope has another name.
Group : SmallGroup(1344,7764)
Rank : 4
Schlafli Type : {12,4,14}
Number of vertices, edges, etc : 12, 24, 28, 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 : {12,2,14}*672, {6,4,14}*672
   3-fold quotients : {4,4,14}*448
   4-fold quotients : {12,2,7}*336, {6,2,14}*336
   6-fold quotients : {2,4,14}*224, {4,2,14}*224
   7-fold quotients : {12,4,2}*192a
   8-fold quotients : {3,2,14}*168, {6,2,7}*168
   12-fold quotients : {4,2,7}*112, {2,2,14}*112
   14-fold quotients : {12,2,2}*96, {6,4,2}*96a
   16-fold quotients : {3,2,7}*84
   21-fold quotients : {4,4,2}*64
   24-fold quotients : {2,2,7}*56
   28-fold quotients : {6,2,2}*48
   42-fold quotients : {2,4,2}*32, {4,2,2}*32
   56-fold quotients : {3,2,2}*24
   84-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  8, 15)(  9, 16)( 10, 17)( 11, 18)( 12, 19)( 13, 20)( 14, 21)( 29, 36)
( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 50, 57)( 51, 58)
( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 71, 78)( 72, 79)( 73, 80)
( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 85,106)( 86,107)( 87,108)( 88,109)
( 89,110)( 90,111)( 91,112)( 92,120)( 93,121)( 94,122)( 95,123)( 96,124)
( 97,125)( 98,126)( 99,113)(100,114)(101,115)(102,116)(103,117)(104,118)
(105,119)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)
(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(141,155)
(142,156)(143,157)(144,158)(145,159)(146,160)(147,161);;
s1 := (  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);;
s2 := (  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)( 85,127)( 86,133)( 87,132)( 88,131)
( 89,130)( 90,129)( 91,128)( 92,134)( 93,140)( 94,139)( 95,138)( 96,137)
( 97,136)( 98,135)( 99,141)(100,147)(101,146)(102,145)(103,144)(104,143)
(105,142)(106,148)(107,154)(108,153)(109,152)(110,151)(111,150)(112,149)
(113,155)(114,161)(115,160)(116,159)(117,158)(118,157)(119,156)(120,162)
(121,168)(122,167)(123,166)(124,165)(125,164)(126,163);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*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, 
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)!(  8, 15)(  9, 16)( 10, 17)( 11, 18)( 12, 19)( 13, 20)( 14, 21)
( 29, 36)( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 50, 57)
( 51, 58)( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 71, 78)( 72, 79)
( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 85,106)( 86,107)( 87,108)
( 88,109)( 89,110)( 90,111)( 91,112)( 92,120)( 93,121)( 94,122)( 95,123)
( 96,124)( 97,125)( 98,126)( 99,113)(100,114)(101,115)(102,116)(103,117)
(104,118)(105,119)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)
(133,154)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)
(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161);
s1 := 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);
s2 := 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)( 85,127)( 86,133)( 87,132)
( 88,131)( 89,130)( 90,129)( 91,128)( 92,134)( 93,140)( 94,139)( 95,138)
( 96,137)( 97,136)( 98,135)( 99,141)(100,147)(101,146)(102,145)(103,144)
(104,143)(105,142)(106,148)(107,154)(108,153)(109,152)(110,151)(111,150)
(112,149)(113,155)(114,161)(115,160)(116,159)(117,158)(118,157)(119,156)
(120,162)(121,168)(122,167)(123,166)(124,165)(125,164)(126,163);
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*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, 
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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