Polytope of Type {6,28,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,28,4}*1344
Also Known As : {{6,28|2},{28,4|2}}. if this polytope has another name.
Group : SmallGroup(1344,7765)
Rank : 4
Schlafli Type : {6,28,4}
Number of vertices, edges, etc : 6, 84, 56, 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 : {6,28,2}*672a, {6,14,4}*672
   3-fold quotients : {2,28,4}*448
   4-fold quotients : {6,14,2}*336
   6-fold quotients : {2,28,2}*224, {2,14,4}*224
   7-fold quotients : {6,4,4}*192
   12-fold quotients : {2,14,2}*112
   14-fold quotients : {6,2,4}*96, {6,4,2}*96a
   21-fold quotients : {2,4,4}*64
   24-fold quotients : {2,7,2}*56
   28-fold quotients : {3,2,4}*48, {6,2,2}*48
   42-fold quotients : {2,2,4}*32, {2,4,2}*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)( 92, 99)( 93,100)( 94,101)( 95,102)
( 96,103)( 97,104)( 98,105)(113,120)(114,121)(115,122)(116,123)(117,124)
(118,125)(119,126)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)
(140,147)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168);;
s1 := (  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);;
s2 := (  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);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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)!(  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)( 92, 99)( 93,100)( 94,101)
( 95,102)( 96,103)( 97,104)( 98,105)(113,120)(114,121)(115,122)(116,123)
(117,124)(118,125)(119,126)(134,141)(135,142)(136,143)(137,144)(138,145)
(139,146)(140,147)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)
(161,168);
s1 := 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);
s2 := 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);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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.
to this polytope