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Polytope of Type {42,6}

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