Polytope of Type {42,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,6,3}*1512b
if this polytope has a name.
Group : SmallGroup(1512,838)
Rank : 4
Schlafli Type : {42,6,3}
Number of vertices, edges, etc : 42, 126, 9, 3
Order of s₀s₁s₂s₃ : 42
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) :
   3-fold quotients : {14,6,3}*504, {42,2,3}*504
   6-fold quotients : {21,2,3}*252
   7-fold quotients : {6,6,3}*216b
   9-fold quotients : {14,2,3}*168
   18-fold quotients : {7,2,3}*84
   21-fold quotients : {2,6,3}*72, {6,2,3}*72
   42-fold quotients : {3,2,3}*36
   63-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  7)(  3,  6)(  4,  5)(  8, 15)(  9, 21)( 10, 20)( 11, 19)( 12, 18)
( 13, 17)( 14, 16)( 23, 28)( 24, 27)( 25, 26)( 29, 36)( 30, 42)( 31, 41)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 44, 49)( 45, 48)( 46, 47)( 50, 57)
( 51, 63)( 52, 62)( 53, 61)( 54, 60)( 55, 59)( 56, 58)( 65, 70)( 66, 69)
( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)( 77, 79)
( 86, 91)( 87, 90)( 88, 89)( 92, 99)( 93,105)( 94,104)( 95,103)( 96,102)
( 97,101)( 98,100)(107,112)(108,111)(109,110)(113,120)(114,126)(115,125)
(116,124)(117,123)(118,122)(119,121)(128,133)(129,132)(130,131)(134,141)
(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(149,154)(150,153)
(151,152)(155,162)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)
(170,175)(171,174)(172,173)(176,183)(177,189)(178,188)(179,187)(180,186)
(181,185)(182,184);;
s1 := (  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)( 15, 16)
( 17, 21)( 18, 20)( 22, 51)( 23, 50)( 24, 56)( 25, 55)( 26, 54)( 27, 53)
( 28, 52)( 29, 44)( 30, 43)( 31, 49)( 32, 48)( 33, 47)( 34, 46)( 35, 45)
( 36, 58)( 37, 57)( 38, 63)( 39, 62)( 40, 61)( 41, 60)( 42, 59)( 64, 72)
( 65, 71)( 66, 77)( 67, 76)( 68, 75)( 69, 74)( 70, 73)( 78, 79)( 80, 84)
( 81, 83)( 85,114)( 86,113)( 87,119)( 88,118)( 89,117)( 90,116)( 91,115)
( 92,107)( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)( 99,121)
(100,120)(101,126)(102,125)(103,124)(104,123)(105,122)(127,135)(128,134)
(129,140)(130,139)(131,138)(132,137)(133,136)(141,142)(143,147)(144,146)
(148,177)(149,176)(150,182)(151,181)(152,180)(153,179)(154,178)(155,170)
(156,169)(157,175)(158,174)(159,173)(160,172)(161,171)(162,184)(163,183)
(164,189)(165,188)(166,187)(167,186)(168,185);;
s2 := (  1, 22)(  2, 23)(  3, 24)(  4, 25)(  5, 26)(  6, 27)(  7, 28)(  8, 29)
(  9, 30)( 10, 31)( 11, 32)( 12, 33)( 13, 34)( 14, 35)( 15, 36)( 16, 37)
( 17, 38)( 18, 39)( 19, 40)( 20, 41)( 21, 42)( 64,148)( 65,149)( 66,150)
( 67,151)( 68,152)( 69,153)( 70,154)( 71,155)( 72,156)( 73,157)( 74,158)
( 75,159)( 76,160)( 77,161)( 78,162)( 79,163)( 80,164)( 81,165)( 82,166)
( 83,167)( 84,168)( 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)(106,169)
(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)
(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,184)(122,185)
(123,186)(124,187)(125,188)(126,189);;
s3 := (  1, 64)(  2, 65)(  3, 66)(  4, 67)(  5, 68)(  6, 69)(  7, 70)(  8, 71)
(  9, 72)( 10, 73)( 11, 74)( 12, 75)( 13, 76)( 14, 77)( 15, 78)( 16, 79)
( 17, 80)( 18, 81)( 19, 82)( 20, 83)( 21, 84)( 22,106)( 23,107)( 24,108)
( 25,109)( 26,110)( 27,111)( 28,112)( 29,113)( 30,114)( 31,115)( 32,116)
( 33,117)( 34,118)( 35,119)( 36,120)( 37,121)( 38,122)( 39,123)( 40,124)
( 41,125)( 42,126)( 43, 85)( 44, 86)( 45, 87)( 46, 88)( 47, 89)( 48, 90)
( 49, 91)( 50, 92)( 51, 93)( 52, 94)( 53, 95)( 54, 96)( 55, 97)( 56, 98)
( 57, 99)( 58,100)( 59,101)( 60,102)( 61,103)( 62,104)( 63,105)(148,169)
(149,170)(150,171)(151,172)(152,173)(153,174)(154,175)(155,176)(156,177)
(157,178)(158,179)(159,180)(160,181)(161,182)(162,183)(163,184)(164,185)
(165,186)(166,187)(167,188)(168,189);;
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, s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(189)!(  2,  7)(  3,  6)(  4,  5)(  8, 15)(  9, 21)( 10, 20)( 11, 19)
( 12, 18)( 13, 17)( 14, 16)( 23, 28)( 24, 27)( 25, 26)( 29, 36)( 30, 42)
( 31, 41)( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 44, 49)( 45, 48)( 46, 47)
( 50, 57)( 51, 63)( 52, 62)( 53, 61)( 54, 60)( 55, 59)( 56, 58)( 65, 70)
( 66, 69)( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)
( 77, 79)( 86, 91)( 87, 90)( 88, 89)( 92, 99)( 93,105)( 94,104)( 95,103)
( 96,102)( 97,101)( 98,100)(107,112)(108,111)(109,110)(113,120)(114,126)
(115,125)(116,124)(117,123)(118,122)(119,121)(128,133)(129,132)(130,131)
(134,141)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(149,154)
(150,153)(151,152)(155,162)(156,168)(157,167)(158,166)(159,165)(160,164)
(161,163)(170,175)(171,174)(172,173)(176,183)(177,189)(178,188)(179,187)
(180,186)(181,185)(182,184);
s1 := Sym(189)!(  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)
( 15, 16)( 17, 21)( 18, 20)( 22, 51)( 23, 50)( 24, 56)( 25, 55)( 26, 54)
( 27, 53)( 28, 52)( 29, 44)( 30, 43)( 31, 49)( 32, 48)( 33, 47)( 34, 46)
( 35, 45)( 36, 58)( 37, 57)( 38, 63)( 39, 62)( 40, 61)( 41, 60)( 42, 59)
( 64, 72)( 65, 71)( 66, 77)( 67, 76)( 68, 75)( 69, 74)( 70, 73)( 78, 79)
( 80, 84)( 81, 83)( 85,114)( 86,113)( 87,119)( 88,118)( 89,117)( 90,116)
( 91,115)( 92,107)( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)
( 99,121)(100,120)(101,126)(102,125)(103,124)(104,123)(105,122)(127,135)
(128,134)(129,140)(130,139)(131,138)(132,137)(133,136)(141,142)(143,147)
(144,146)(148,177)(149,176)(150,182)(151,181)(152,180)(153,179)(154,178)
(155,170)(156,169)(157,175)(158,174)(159,173)(160,172)(161,171)(162,184)
(163,183)(164,189)(165,188)(166,187)(167,186)(168,185);
s2 := Sym(189)!(  1, 22)(  2, 23)(  3, 24)(  4, 25)(  5, 26)(  6, 27)(  7, 28)
(  8, 29)(  9, 30)( 10, 31)( 11, 32)( 12, 33)( 13, 34)( 14, 35)( 15, 36)
( 16, 37)( 17, 38)( 18, 39)( 19, 40)( 20, 41)( 21, 42)( 64,148)( 65,149)
( 66,150)( 67,151)( 68,152)( 69,153)( 70,154)( 71,155)( 72,156)( 73,157)
( 74,158)( 75,159)( 76,160)( 77,161)( 78,162)( 79,163)( 80,164)( 81,165)
( 82,166)( 83,167)( 84,168)( 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)
(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)
(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,184)
(122,185)(123,186)(124,187)(125,188)(126,189);
s3 := Sym(189)!(  1, 64)(  2, 65)(  3, 66)(  4, 67)(  5, 68)(  6, 69)(  7, 70)
(  8, 71)(  9, 72)( 10, 73)( 11, 74)( 12, 75)( 13, 76)( 14, 77)( 15, 78)
( 16, 79)( 17, 80)( 18, 81)( 19, 82)( 20, 83)( 21, 84)( 22,106)( 23,107)
( 24,108)( 25,109)( 26,110)( 27,111)( 28,112)( 29,113)( 30,114)( 31,115)
( 32,116)( 33,117)( 34,118)( 35,119)( 36,120)( 37,121)( 38,122)( 39,123)
( 40,124)( 41,125)( 42,126)( 43, 85)( 44, 86)( 45, 87)( 46, 88)( 47, 89)
( 48, 90)( 49, 91)( 50, 92)( 51, 93)( 52, 94)( 53, 95)( 54, 96)( 55, 97)
( 56, 98)( 57, 99)( 58,100)( 59,101)( 60,102)( 61,103)( 62,104)( 63,105)
(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)(154,175)(155,176)
(156,177)(157,178)(158,179)(159,180)(160,181)(161,182)(162,183)(163,184)
(164,185)(165,186)(166,187)(167,188)(168,189);
poly := sub<Sym(189)|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, 
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope