Polytope of Type {6,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,42}*672
if this polytope has a name.
Group : SmallGroup(672,1260)
Rank : 3
Schlafli Type : {6,42}
Number of vertices, edges, etc : 8, 168, 56
Order of s₀s₁s₂ : 28
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,42,2} of size 1344
Vertex Figure Of :
   {2,6,42} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,21}*336
   7-fold quotients : {6,6}*96
   12-fold quotients : {2,14}*56
   14-fold quotients : {3,6}*48, {6,3}*48
   24-fold quotients : {2,7}*28
   28-fold quotients : {3,3}*24
   84-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,84}*1344a, {12,42}*1344a, {6,42}*1344, {6,84}*1344b, {12,42}*1344b
Permutation Representation (GAP) :

s0 := (  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 22, 23)( 26, 27)( 29, 57)
( 30, 59)( 31, 58)( 32, 60)( 33, 61)( 34, 63)( 35, 62)( 36, 64)( 37, 65)
( 38, 67)( 39, 66)( 40, 68)( 41, 69)( 42, 71)( 43, 70)( 44, 72)( 45, 73)
( 46, 75)( 47, 74)( 48, 76)( 49, 77)( 50, 79)( 51, 78)( 52, 80)( 53, 81)
( 54, 83)( 55, 82)( 56, 84)( 86, 87)( 90, 91)( 94, 95)( 98, 99)(102,103)
(106,107)(110,111)(113,141)(114,143)(115,142)(116,144)(117,145)(118,147)
(119,146)(120,148)(121,149)(122,151)(123,150)(124,152)(125,153)(126,155)
(127,154)(128,156)(129,157)(130,159)(131,158)(132,160)(133,161)(134,163)
(135,162)(136,164)(137,165)(138,167)(139,166)(140,168);;
s1 := (  1, 29)(  2, 30)(  3, 32)(  4, 31)(  5, 53)(  6, 54)(  7, 56)(  8, 55)
(  9, 49)( 10, 50)( 11, 52)( 12, 51)( 13, 45)( 14, 46)( 15, 48)( 16, 47)
( 17, 41)( 18, 42)( 19, 44)( 20, 43)( 21, 37)( 22, 38)( 23, 40)( 24, 39)
( 25, 33)( 26, 34)( 27, 36)( 28, 35)( 59, 60)( 61, 81)( 62, 82)( 63, 84)
( 64, 83)( 65, 77)( 66, 78)( 67, 80)( 68, 79)( 69, 73)( 70, 74)( 71, 76)
( 72, 75)( 85,113)( 86,114)( 87,116)( 88,115)( 89,137)( 90,138)( 91,140)
( 92,139)( 93,133)( 94,134)( 95,136)( 96,135)( 97,129)( 98,130)( 99,132)
(100,131)(101,125)(102,126)(103,128)(104,127)(105,121)(106,122)(107,124)
(108,123)(109,117)(110,118)(111,120)(112,119)(143,144)(145,165)(146,166)
(147,168)(148,167)(149,161)(150,162)(151,164)(152,163)(153,157)(154,158)
(155,160)(156,159);;
s2 := (  1, 92)(  2, 90)(  3, 91)(  4, 89)(  5, 88)(  6, 86)(  7, 87)(  8, 85)
(  9,112)( 10,110)( 11,111)( 12,109)( 13,108)( 14,106)( 15,107)( 16,105)
( 17,104)( 18,102)( 19,103)( 20,101)( 21,100)( 22, 98)( 23, 99)( 24, 97)
( 25, 96)( 26, 94)( 27, 95)( 28, 93)( 29,148)( 30,146)( 31,147)( 32,145)
( 33,144)( 34,142)( 35,143)( 36,141)( 37,168)( 38,166)( 39,167)( 40,165)
( 41,164)( 42,162)( 43,163)( 44,161)( 45,160)( 46,158)( 47,159)( 48,157)
( 49,156)( 50,154)( 51,155)( 52,153)( 53,152)( 54,150)( 55,151)( 56,149)
( 57,120)( 58,118)( 59,119)( 60,117)( 61,116)( 62,114)( 63,115)( 64,113)
( 65,140)( 66,138)( 67,139)( 68,137)( 69,136)( 70,134)( 71,135)( 72,133)
( 73,132)( 74,130)( 75,131)( 76,129)( 77,128)( 78,126)( 79,127)( 80,125)
( 81,124)( 82,122)( 83,123)( 84,121);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(168)!(  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 22, 23)( 26, 27)
( 29, 57)( 30, 59)( 31, 58)( 32, 60)( 33, 61)( 34, 63)( 35, 62)( 36, 64)
( 37, 65)( 38, 67)( 39, 66)( 40, 68)( 41, 69)( 42, 71)( 43, 70)( 44, 72)
( 45, 73)( 46, 75)( 47, 74)( 48, 76)( 49, 77)( 50, 79)( 51, 78)( 52, 80)
( 53, 81)( 54, 83)( 55, 82)( 56, 84)( 86, 87)( 90, 91)( 94, 95)( 98, 99)
(102,103)(106,107)(110,111)(113,141)(114,143)(115,142)(116,144)(117,145)
(118,147)(119,146)(120,148)(121,149)(122,151)(123,150)(124,152)(125,153)
(126,155)(127,154)(128,156)(129,157)(130,159)(131,158)(132,160)(133,161)
(134,163)(135,162)(136,164)(137,165)(138,167)(139,166)(140,168);
s1 := Sym(168)!(  1, 29)(  2, 30)(  3, 32)(  4, 31)(  5, 53)(  6, 54)(  7, 56)
(  8, 55)(  9, 49)( 10, 50)( 11, 52)( 12, 51)( 13, 45)( 14, 46)( 15, 48)
( 16, 47)( 17, 41)( 18, 42)( 19, 44)( 20, 43)( 21, 37)( 22, 38)( 23, 40)
( 24, 39)( 25, 33)( 26, 34)( 27, 36)( 28, 35)( 59, 60)( 61, 81)( 62, 82)
( 63, 84)( 64, 83)( 65, 77)( 66, 78)( 67, 80)( 68, 79)( 69, 73)( 70, 74)
( 71, 76)( 72, 75)( 85,113)( 86,114)( 87,116)( 88,115)( 89,137)( 90,138)
( 91,140)( 92,139)( 93,133)( 94,134)( 95,136)( 96,135)( 97,129)( 98,130)
( 99,132)(100,131)(101,125)(102,126)(103,128)(104,127)(105,121)(106,122)
(107,124)(108,123)(109,117)(110,118)(111,120)(112,119)(143,144)(145,165)
(146,166)(147,168)(148,167)(149,161)(150,162)(151,164)(152,163)(153,157)
(154,158)(155,160)(156,159);
s2 := Sym(168)!(  1, 92)(  2, 90)(  3, 91)(  4, 89)(  5, 88)(  6, 86)(  7, 87)
(  8, 85)(  9,112)( 10,110)( 11,111)( 12,109)( 13,108)( 14,106)( 15,107)
( 16,105)( 17,104)( 18,102)( 19,103)( 20,101)( 21,100)( 22, 98)( 23, 99)
( 24, 97)( 25, 96)( 26, 94)( 27, 95)( 28, 93)( 29,148)( 30,146)( 31,147)
( 32,145)( 33,144)( 34,142)( 35,143)( 36,141)( 37,168)( 38,166)( 39,167)
( 40,165)( 41,164)( 42,162)( 43,163)( 44,161)( 45,160)( 46,158)( 47,159)
( 48,157)( 49,156)( 50,154)( 51,155)( 52,153)( 53,152)( 54,150)( 55,151)
( 56,149)( 57,120)( 58,118)( 59,119)( 60,117)( 61,116)( 62,114)( 63,115)
( 64,113)( 65,140)( 66,138)( 67,139)( 68,137)( 69,136)( 70,134)( 71,135)
( 72,133)( 73,132)( 74,130)( 75,131)( 76,129)( 77,128)( 78,126)( 79,127)
( 80,125)( 81,124)( 82,122)( 83,123)( 84,121);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2 >;

References : None.
to this polytope