Polytope of Type {6,4,28}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,28}*1344
Also Known As : {{6,4|2},{4,28|2}}. if this polytope has another name.
Group : SmallGroup(1344,7765)
Rank : 4
Schlafli Type : {6,4,28}
Number of vertices, edges, etc : 6, 12, 56, 28
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 : {6,2,28}*672, {6,4,14}*672
   3-fold quotients : {2,4,28}*448
   4-fold quotients : {3,2,28}*336, {6,2,14}*336
   6-fold quotients : {2,2,28}*224, {2,4,14}*224
   7-fold quotients : {6,4,4}*192
   8-fold quotients : {3,2,14}*168, {6,2,7}*168
   12-fold quotients : {2,2,14}*112
   14-fold quotients : {6,2,4}*96, {6,4,2}*96a
   16-fold quotients : {3,2,7}*84
   21-fold quotients : {2,4,4}*64
   24-fold quotients : {2,2,7}*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,  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)( 85,134)( 86,135)( 87,136)( 88,137)
( 89,138)( 90,139)( 91,140)( 92,127)( 93,128)( 94,129)( 95,130)( 96,131)
( 97,132)( 98,133)( 99,141)(100,142)(101,143)(102,144)(103,145)(104,146)
(105,147)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,161)
(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,162)
(121,163)(122,164)(123,165)(124,166)(125,167)(126,168);;
s2 := (  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);;
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)( 85,107)( 86,106)( 87,112)( 88,111)
( 89,110)( 90,109)( 91,108)( 92,114)( 93,113)( 94,119)( 95,118)( 96,117)
( 97,116)( 98,115)( 99,121)(100,120)(101,126)(102,125)(103,124)(104,123)
(105,122)(127,149)(128,148)(129,154)(130,153)(131,152)(132,151)(133,150)
(134,156)(135,155)(136,161)(137,160)(138,159)(139,158)(140,157)(141,163)
(142,162)(143,168)(144,167)(145,166)(146,165)(147,164);;
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, 
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*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)( 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,  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)( 85,134)( 86,135)( 87,136)
( 88,137)( 89,138)( 90,139)( 91,140)( 92,127)( 93,128)( 94,129)( 95,130)
( 96,131)( 97,132)( 98,133)( 99,141)(100,142)(101,143)(102,144)(103,145)
(104,146)(105,147)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)
(112,161)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)
(120,162)(121,163)(122,164)(123,165)(124,166)(125,167)(126,168);
s2 := 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);
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)( 85,107)( 86,106)( 87,112)
( 88,111)( 89,110)( 90,109)( 91,108)( 92,114)( 93,113)( 94,119)( 95,118)
( 96,117)( 97,116)( 98,115)( 99,121)(100,120)(101,126)(102,125)(103,124)
(104,123)(105,122)(127,149)(128,148)(129,154)(130,153)(131,152)(132,151)
(133,150)(134,156)(135,155)(136,161)(137,160)(138,159)(139,158)(140,157)
(141,163)(142,162)(143,168)(144,167)(145,166)(146,165)(147,164);
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, 
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*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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