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Polytope of Type {28,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,6}*672
if this polytope has a name.
Group : SmallGroup(672,1260)
Rank : 3
Schlafli Type : {28,6}
Number of vertices, edges, etc : 56, 168, 12
Order of s0s1s2 : 42
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {28,6,2} of size 1344
Vertex Figure Of :
   {2,28,6} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {28,6}*336b
   4-fold quotients : {14,6}*168
   7-fold quotients : {4,6}*96
   12-fold quotients : {14,2}*56
   14-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   24-fold quotients : {7,2}*28
   28-fold quotients : {4,3}*24, {2,6}*24
   56-fold quotients : {2,3}*12
   84-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {28,12}*1344b, {28,6}*1344e, {56,6}*1344b, {56,6}*1344c, {28,12}*1344c
Permutation Representation (GAP) :
s0 := (  1,  3)(  2,  4)(  5, 27)(  6, 28)(  7, 25)(  8, 26)(  9, 23)( 10, 24)
( 11, 21)( 12, 22)( 13, 19)( 14, 20)( 15, 17)( 16, 18)( 29, 31)( 30, 32)
( 33, 55)( 34, 56)( 35, 53)( 36, 54)( 37, 51)( 38, 52)( 39, 49)( 40, 50)
( 41, 47)( 42, 48)( 43, 45)( 44, 46)( 57, 59)( 58, 60)( 61, 83)( 62, 84)
( 63, 81)( 64, 82)( 65, 79)( 66, 80)( 67, 77)( 68, 78)( 69, 75)( 70, 76)
( 71, 73)( 72, 74)( 85, 87)( 86, 88)( 89,111)( 90,112)( 91,109)( 92,110)
( 93,107)( 94,108)( 95,105)( 96,106)( 97,103)( 98,104)( 99,101)(100,102)
(113,115)(114,116)(117,139)(118,140)(119,137)(120,138)(121,135)(122,136)
(123,133)(124,134)(125,131)(126,132)(127,129)(128,130)(141,143)(142,144)
(145,167)(146,168)(147,165)(148,166)(149,163)(150,164)(151,161)(152,162)
(153,159)(154,160)(155,157)(156,158);;
s1 := (  1,  5)(  2,  7)(  3,  6)(  4,  8)(  9, 25)( 10, 27)( 11, 26)( 12, 28)
( 13, 21)( 14, 23)( 15, 22)( 16, 24)( 18, 19)( 29, 61)( 30, 63)( 31, 62)
( 32, 64)( 33, 57)( 34, 59)( 35, 58)( 36, 60)( 37, 81)( 38, 83)( 39, 82)
( 40, 84)( 41, 77)( 42, 79)( 43, 78)( 44, 80)( 45, 73)( 46, 75)( 47, 74)
( 48, 76)( 49, 69)( 50, 71)( 51, 70)( 52, 72)( 53, 65)( 54, 67)( 55, 66)
( 56, 68)( 85, 89)( 86, 91)( 87, 90)( 88, 92)( 93,109)( 94,111)( 95,110)
( 96,112)( 97,105)( 98,107)( 99,106)(100,108)(102,103)(113,145)(114,147)
(115,146)(116,148)(117,141)(118,143)(119,142)(120,144)(121,165)(122,167)
(123,166)(124,168)(125,161)(126,163)(127,162)(128,164)(129,157)(130,159)
(131,158)(132,160)(133,153)(134,155)(135,154)(136,156)(137,149)(138,151)
(139,150)(140,152);;
s2 := (  1,141)(  2,144)(  3,143)(  4,142)(  5,145)(  6,148)(  7,147)(  8,146)
(  9,149)( 10,152)( 11,151)( 12,150)( 13,153)( 14,156)( 15,155)( 16,154)
( 17,157)( 18,160)( 19,159)( 20,158)( 21,161)( 22,164)( 23,163)( 24,162)
( 25,165)( 26,168)( 27,167)( 28,166)( 29,113)( 30,116)( 31,115)( 32,114)
( 33,117)( 34,120)( 35,119)( 36,118)( 37,121)( 38,124)( 39,123)( 40,122)
( 41,125)( 42,128)( 43,127)( 44,126)( 45,129)( 46,132)( 47,131)( 48,130)
( 49,133)( 50,136)( 51,135)( 52,134)( 53,137)( 54,140)( 55,139)( 56,138)
( 57, 85)( 58, 88)( 59, 87)( 60, 86)( 61, 89)( 62, 92)( 63, 91)( 64, 90)
( 65, 93)( 66, 96)( 67, 95)( 68, 94)( 69, 97)( 70,100)( 71, 99)( 72, 98)
( 73,101)( 74,104)( 75,103)( 76,102)( 77,105)( 78,108)( 79,107)( 80,106)
( 81,109)( 82,112)( 83,111)( 84,110);;
poly := Group([s0,s1,s2]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(168)!(  1,  3)(  2,  4)(  5, 27)(  6, 28)(  7, 25)(  8, 26)(  9, 23)
( 10, 24)( 11, 21)( 12, 22)( 13, 19)( 14, 20)( 15, 17)( 16, 18)( 29, 31)
( 30, 32)( 33, 55)( 34, 56)( 35, 53)( 36, 54)( 37, 51)( 38, 52)( 39, 49)
( 40, 50)( 41, 47)( 42, 48)( 43, 45)( 44, 46)( 57, 59)( 58, 60)( 61, 83)
( 62, 84)( 63, 81)( 64, 82)( 65, 79)( 66, 80)( 67, 77)( 68, 78)( 69, 75)
( 70, 76)( 71, 73)( 72, 74)( 85, 87)( 86, 88)( 89,111)( 90,112)( 91,109)
( 92,110)( 93,107)( 94,108)( 95,105)( 96,106)( 97,103)( 98,104)( 99,101)
(100,102)(113,115)(114,116)(117,139)(118,140)(119,137)(120,138)(121,135)
(122,136)(123,133)(124,134)(125,131)(126,132)(127,129)(128,130)(141,143)
(142,144)(145,167)(146,168)(147,165)(148,166)(149,163)(150,164)(151,161)
(152,162)(153,159)(154,160)(155,157)(156,158);
s1 := Sym(168)!(  1,  5)(  2,  7)(  3,  6)(  4,  8)(  9, 25)( 10, 27)( 11, 26)
( 12, 28)( 13, 21)( 14, 23)( 15, 22)( 16, 24)( 18, 19)( 29, 61)( 30, 63)
( 31, 62)( 32, 64)( 33, 57)( 34, 59)( 35, 58)( 36, 60)( 37, 81)( 38, 83)
( 39, 82)( 40, 84)( 41, 77)( 42, 79)( 43, 78)( 44, 80)( 45, 73)( 46, 75)
( 47, 74)( 48, 76)( 49, 69)( 50, 71)( 51, 70)( 52, 72)( 53, 65)( 54, 67)
( 55, 66)( 56, 68)( 85, 89)( 86, 91)( 87, 90)( 88, 92)( 93,109)( 94,111)
( 95,110)( 96,112)( 97,105)( 98,107)( 99,106)(100,108)(102,103)(113,145)
(114,147)(115,146)(116,148)(117,141)(118,143)(119,142)(120,144)(121,165)
(122,167)(123,166)(124,168)(125,161)(126,163)(127,162)(128,164)(129,157)
(130,159)(131,158)(132,160)(133,153)(134,155)(135,154)(136,156)(137,149)
(138,151)(139,150)(140,152);
s2 := Sym(168)!(  1,141)(  2,144)(  3,143)(  4,142)(  5,145)(  6,148)(  7,147)
(  8,146)(  9,149)( 10,152)( 11,151)( 12,150)( 13,153)( 14,156)( 15,155)
( 16,154)( 17,157)( 18,160)( 19,159)( 20,158)( 21,161)( 22,164)( 23,163)
( 24,162)( 25,165)( 26,168)( 27,167)( 28,166)( 29,113)( 30,116)( 31,115)
( 32,114)( 33,117)( 34,120)( 35,119)( 36,118)( 37,121)( 38,124)( 39,123)
( 40,122)( 41,125)( 42,128)( 43,127)( 44,126)( 45,129)( 46,132)( 47,131)
( 48,130)( 49,133)( 50,136)( 51,135)( 52,134)( 53,137)( 54,140)( 55,139)
( 56,138)( 57, 85)( 58, 88)( 59, 87)( 60, 86)( 61, 89)( 62, 92)( 63, 91)
( 64, 90)( 65, 93)( 66, 96)( 67, 95)( 68, 94)( 69, 97)( 70,100)( 71, 99)
( 72, 98)( 73,101)( 74,104)( 75,103)( 76,102)( 77,105)( 78,108)( 79,107)
( 80,106)( 81,109)( 82,112)( 83,111)( 84,110);
poly := sub<Sym(168)|s0,s1,s2>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2 >; 
 
References : None.
to this polytope