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Polytope of Type {4,28,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,28,6}*1344
Also Known As : {{4,28|2},{28,6|2}}. if this polytope has another name.
Group : SmallGroup(1344,7765)
Rank : 4
Schlafli Type : {4,28,6}
Number of vertices, edges, etc : 4, 56, 84, 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 : {2,28,6}*672a, {4,14,6}*672
   3-fold quotients : {4,28,2}*448
   4-fold quotients : {2,14,6}*336
   6-fold quotients : {2,28,2}*224, {4,14,2}*224
   7-fold quotients : {4,4,6}*192
   12-fold quotients : {2,14,2}*112
   14-fold quotients : {2,4,6}*96a, {4,2,6}*96
   21-fold quotients : {4,4,2}*64
   24-fold quotients : {2,7,2}*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 := ( 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, 91)(  3, 90)(  4, 89)(  5, 88)(  6, 87)(  7, 86)(  8, 92)
(  9, 98)( 10, 97)( 11, 96)( 12, 95)( 13, 94)( 14, 93)( 15, 99)( 16,105)
( 17,104)( 18,103)( 19,102)( 20,101)( 21,100)( 22,106)( 23,112)( 24,111)
( 25,110)( 26,109)( 27,108)( 28,107)( 29,113)( 30,119)( 31,118)( 32,117)
( 33,116)( 34,115)( 35,114)( 36,120)( 37,126)( 38,125)( 39,124)( 40,123)
( 41,122)( 42,121)( 43,127)( 44,133)( 45,132)( 46,131)( 47,130)( 48,129)
( 49,128)( 50,134)( 51,140)( 52,139)( 53,138)( 54,137)( 55,136)( 56,135)
( 57,141)( 58,147)( 59,146)( 60,145)( 61,144)( 62,143)( 63,142)( 64,148)
( 65,154)( 66,153)( 67,152)( 68,151)( 69,150)( 70,149)( 71,155)( 72,161)
( 73,160)( 74,159)( 75,158)( 76,157)( 77,156)( 78,162)( 79,168)( 80,167)
( 81,166)( 82,165)( 83,164)( 84,163);;
s2 := (  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);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*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*s1*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*s1*s2*s1*s2*s1*s2 ];;
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, 91)(  3, 90)(  4, 89)(  5, 88)(  6, 87)(  7, 86)
(  8, 92)(  9, 98)( 10, 97)( 11, 96)( 12, 95)( 13, 94)( 14, 93)( 15, 99)
( 16,105)( 17,104)( 18,103)( 19,102)( 20,101)( 21,100)( 22,106)( 23,112)
( 24,111)( 25,110)( 26,109)( 27,108)( 28,107)( 29,113)( 30,119)( 31,118)
( 32,117)( 33,116)( 34,115)( 35,114)( 36,120)( 37,126)( 38,125)( 39,124)
( 40,123)( 41,122)( 42,121)( 43,127)( 44,133)( 45,132)( 46,131)( 47,130)
( 48,129)( 49,128)( 50,134)( 51,140)( 52,139)( 53,138)( 54,137)( 55,136)
( 56,135)( 57,141)( 58,147)( 59,146)( 60,145)( 61,144)( 62,143)( 63,142)
( 64,148)( 65,154)( 66,153)( 67,152)( 68,151)( 69,150)( 70,149)( 71,155)
( 72,161)( 73,160)( 74,159)( 75,158)( 76,157)( 77,156)( 78,162)( 79,168)
( 80,167)( 81,166)( 82,165)( 83,164)( 84,163);
s2 := 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);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*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*s1*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*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
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