Polytope of Type {6,21,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,21,6}*1512
if this polytope has a name.
Group : SmallGroup(1512,838)
Rank : 4
Schlafli Type : {6,21,6}
Number of vertices, edges, etc : 6, 63, 63, 6
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
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 : {2,21,6}*504, {6,21,2}*504
   7-fold quotients : {6,3,6}*216
   9-fold quotients : {2,21,2}*168
   21-fold quotients : {2,3,6}*72, {6,3,2}*72
   27-fold quotients : {2,7,2}*56
   63-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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