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Polytope of Type {28,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,4,6}*1344
Also Known As : {{28,4|2},{4,6|2}}. if this polytope has another name.
Group : SmallGroup(1344,7765)
Rank : 4
Schlafli Type : {28,4,6}
Number of vertices, edges, etc : 28, 56, 12, 6
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 : {28,2,6}*672, {14,4,6}*672
   3-fold quotients : {28,4,2}*448
   4-fold quotients : {28,2,3}*336, {14,2,6}*336
   6-fold quotients : {28,2,2}*224, {14,4,2}*224
   7-fold quotients : {4,4,6}*192
   8-fold quotients : {7,2,6}*168, {14,2,3}*168
   12-fold quotients : {14,2,2}*112
   14-fold quotients : {2,4,6}*96a, {4,2,6}*96
   16-fold quotients : {7,2,3}*84
   21-fold quotients : {4,4,2}*64
   24-fold quotients : {7,2,2}*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 := (  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)( 85,106)( 86,112)( 87,111)( 88,110)
( 89,109)( 90,108)( 91,107)( 92,113)( 93,119)( 94,118)( 95,117)( 96,116)
( 97,115)( 98,114)( 99,120)(100,126)(101,125)(102,124)(103,123)(104,122)
(105,121)(127,148)(128,154)(129,153)(130,152)(131,151)(132,150)(133,149)
(134,155)(135,161)(136,160)(137,159)(138,158)(139,157)(140,156)(141,162)
(142,168)(143,167)(144,166)(145,165)(146,164)(147,163);;
s1 := (  1, 86)(  2, 85)(  3, 91)(  4, 90)(  5, 89)(  6, 88)(  7, 87)(  8, 93)
(  9, 92)( 10, 98)( 11, 97)( 12, 96)( 13, 95)( 14, 94)( 15,100)( 16, 99)
( 17,105)( 18,104)( 19,103)( 20,102)( 21,101)( 22,107)( 23,106)( 24,112)
( 25,111)( 26,110)( 27,109)( 28,108)( 29,114)( 30,113)( 31,119)( 32,118)
( 33,117)( 34,116)( 35,115)( 36,121)( 37,120)( 38,126)( 39,125)( 40,124)
( 41,123)( 42,122)( 43,128)( 44,127)( 45,133)( 46,132)( 47,131)( 48,130)
( 49,129)( 50,135)( 51,134)( 52,140)( 53,139)( 54,138)( 55,137)( 56,136)
( 57,142)( 58,141)( 59,147)( 60,146)( 61,145)( 62,144)( 63,143)( 64,149)
( 65,148)( 66,154)( 67,153)( 68,152)( 69,151)( 70,150)( 71,156)( 72,155)
( 73,161)( 74,160)( 75,159)( 76,158)( 77,157)( 78,163)( 79,162)( 80,168)
( 81,167)( 82,166)( 83,165)( 84,164);;
s2 := (  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)( 85,127)( 86,128)( 87,129)( 88,130)
( 89,131)( 90,132)( 91,133)( 92,141)( 93,142)( 94,143)( 95,144)( 96,145)
( 97,146)( 98,147)( 99,134)(100,135)(101,136)(102,137)(103,138)(104,139)
(105,140)(106,148)(107,149)(108,150)(109,151)(110,152)(111,153)(112,154)
(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,155)
(121,156)(122,157)(123,158)(124,159)(125,160)(126,161);;
s3 := (  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)( 22, 29)
( 23, 30)( 24, 31)( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 43, 50)( 44, 51)
( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)( 66, 73)
( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85, 92)( 86, 93)( 87, 94)( 88, 95)
( 89, 96)( 90, 97)( 91, 98)(106,113)(107,114)(108,115)(109,116)(110,117)
(111,118)(112,119)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)
(133,140)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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)(  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)( 85,106)( 86,112)( 87,111)
( 88,110)( 89,109)( 90,108)( 91,107)( 92,113)( 93,119)( 94,118)( 95,117)
( 96,116)( 97,115)( 98,114)( 99,120)(100,126)(101,125)(102,124)(103,123)
(104,122)(105,121)(127,148)(128,154)(129,153)(130,152)(131,151)(132,150)
(133,149)(134,155)(135,161)(136,160)(137,159)(138,158)(139,157)(140,156)
(141,162)(142,168)(143,167)(144,166)(145,165)(146,164)(147,163);
s1 := Sym(168)!(  1, 86)(  2, 85)(  3, 91)(  4, 90)(  5, 89)(  6, 88)(  7, 87)
(  8, 93)(  9, 92)( 10, 98)( 11, 97)( 12, 96)( 13, 95)( 14, 94)( 15,100)
( 16, 99)( 17,105)( 18,104)( 19,103)( 20,102)( 21,101)( 22,107)( 23,106)
( 24,112)( 25,111)( 26,110)( 27,109)( 28,108)( 29,114)( 30,113)( 31,119)
( 32,118)( 33,117)( 34,116)( 35,115)( 36,121)( 37,120)( 38,126)( 39,125)
( 40,124)( 41,123)( 42,122)( 43,128)( 44,127)( 45,133)( 46,132)( 47,131)
( 48,130)( 49,129)( 50,135)( 51,134)( 52,140)( 53,139)( 54,138)( 55,137)
( 56,136)( 57,142)( 58,141)( 59,147)( 60,146)( 61,145)( 62,144)( 63,143)
( 64,149)( 65,148)( 66,154)( 67,153)( 68,152)( 69,151)( 70,150)( 71,156)
( 72,155)( 73,161)( 74,160)( 75,159)( 76,158)( 77,157)( 78,163)( 79,162)
( 80,168)( 81,167)( 82,166)( 83,165)( 84,164);
s2 := 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)( 85,127)( 86,128)( 87,129)
( 88,130)( 89,131)( 90,132)( 91,133)( 92,141)( 93,142)( 94,143)( 95,144)
( 96,145)( 97,146)( 98,147)( 99,134)(100,135)(101,136)(102,137)(103,138)
(104,139)(105,140)(106,148)(107,149)(108,150)(109,151)(110,152)(111,153)
(112,154)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)
(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161);
s3 := Sym(168)!(  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)
( 22, 29)( 23, 30)( 24, 31)( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 43, 50)
( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)
( 66, 73)( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85, 92)( 86, 93)( 87, 94)
( 88, 95)( 89, 96)( 90, 97)( 91, 98)(106,113)(107,114)(108,115)(109,116)
(110,117)(111,118)(112,119)(127,134)(128,135)(129,136)(130,137)(131,138)
(132,139)(133,140)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)
(154,161);
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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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