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Polytope of Type {21,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {21,6,6}*1512b
if this polytope has a name.
Group : SmallGroup(1512,838)
Rank : 4
Schlafli Type : {21,6,6}
Number of vertices, edges, etc : 21, 63, 18, 6
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 : {21,2,6}*504, {21,6,2}*504
   6-fold quotients : {21,2,3}*252
   7-fold quotients : {3,6,6}*216b
   9-fold quotients : {7,2,6}*168, {21,2,2}*168
   18-fold quotients : {7,2,3}*84
   21-fold quotients : {3,2,6}*72, {3,6,2}*72
   27-fold quotients : {7,2,2}*56
   42-fold quotients : {3,2,3}*36
   63-fold quotients : {3,2,2}*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)( 22, 43)( 23, 49)( 24, 48)( 25, 47)( 26, 46)( 27, 45)
( 28, 44)( 29, 57)( 30, 63)( 31, 62)( 32, 61)( 33, 60)( 34, 59)( 35, 58)
( 36, 50)( 37, 56)( 38, 55)( 39, 54)( 40, 53)( 41, 52)( 42, 51)( 65, 70)
( 66, 69)( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)
( 77, 79)( 85,106)( 86,112)( 87,111)( 88,110)( 89,109)( 90,108)( 91,107)
( 92,120)( 93,126)( 94,125)( 95,124)( 96,123)( 97,122)( 98,121)( 99,113)
(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(128,133)(129,132)
(130,131)(134,141)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)
(148,169)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,183)
(156,189)(157,188)(158,187)(159,186)(160,185)(161,184)(162,176)(163,182)
(164,181)(165,180)(166,179)(167,178)(168,177);;
s1 := (  1, 30)(  2, 29)(  3, 35)(  4, 34)(  5, 33)(  6, 32)(  7, 31)(  8, 23)
(  9, 22)( 10, 28)( 11, 27)( 12, 26)( 13, 25)( 14, 24)( 15, 37)( 16, 36)
( 17, 42)( 18, 41)( 19, 40)( 20, 39)( 21, 38)( 43, 51)( 44, 50)( 45, 56)
( 46, 55)( 47, 54)( 48, 53)( 49, 52)( 57, 58)( 59, 63)( 60, 62)( 64, 93)
( 65, 92)( 66, 98)( 67, 97)( 68, 96)( 69, 95)( 70, 94)( 71, 86)( 72, 85)
( 73, 91)( 74, 90)( 75, 89)( 76, 88)( 77, 87)( 78,100)( 79, 99)( 80,105)
( 81,104)( 82,103)( 83,102)( 84,101)(106,114)(107,113)(108,119)(109,118)
(110,117)(111,116)(112,115)(120,121)(122,126)(123,125)(127,156)(128,155)
(129,161)(130,160)(131,159)(132,158)(133,157)(134,149)(135,148)(136,154)
(137,153)(138,152)(139,151)(140,150)(141,163)(142,162)(143,168)(144,167)
(145,166)(146,165)(147,164)(169,177)(170,176)(171,182)(172,181)(173,180)
(174,179)(175,178)(183,184)(185,189)(186,188);;
s2 := ( 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)( 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)( 85,169)( 86,170)( 87,171)( 88,172)( 89,173)( 90,174)
( 91,175)( 92,176)( 93,177)( 94,178)( 95,179)( 96,180)( 97,181)( 98,182)
( 99,183)(100,184)(101,185)(102,186)(103,187)(104,188)(105,189)(106,148)
(107,149)(108,150)(109,151)(110,152)(111,153)(112,154)(113,155)(114,156)
(115,157)(116,158)(117,159)(118,160)(119,161)(120,162)(121,163)(122,164)
(123,165)(124,166)(125,167)(126,168);;
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, 85)( 23, 86)( 24, 87)
( 25, 88)( 26, 89)( 27, 90)( 28, 91)( 29, 92)( 30, 93)( 31, 94)( 32, 95)
( 33, 96)( 34, 97)( 35, 98)( 36, 99)( 37,100)( 38,101)( 39,102)( 40,103)
( 41,104)( 42,105)( 43,106)( 44,107)( 45,108)( 46,109)( 47,110)( 48,111)
( 49,112)( 50,113)( 51,114)( 52,115)( 53,116)( 54,117)( 55,118)( 56,119)
( 57,120)( 58,121)( 59,122)( 60,123)( 61,124)( 62,125)( 63,126);;
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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)( 22, 43)( 23, 49)( 24, 48)( 25, 47)( 26, 46)
( 27, 45)( 28, 44)( 29, 57)( 30, 63)( 31, 62)( 32, 61)( 33, 60)( 34, 59)
( 35, 58)( 36, 50)( 37, 56)( 38, 55)( 39, 54)( 40, 53)( 41, 52)( 42, 51)
( 65, 70)( 66, 69)( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)
( 76, 80)( 77, 79)( 85,106)( 86,112)( 87,111)( 88,110)( 89,109)( 90,108)
( 91,107)( 92,120)( 93,126)( 94,125)( 95,124)( 96,123)( 97,122)( 98,121)
( 99,113)(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(128,133)
(129,132)(130,131)(134,141)(135,147)(136,146)(137,145)(138,144)(139,143)
(140,142)(148,169)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)
(155,183)(156,189)(157,188)(158,187)(159,186)(160,185)(161,184)(162,176)
(163,182)(164,181)(165,180)(166,179)(167,178)(168,177);
s1 := Sym(189)!(  1, 30)(  2, 29)(  3, 35)(  4, 34)(  5, 33)(  6, 32)(  7, 31)
(  8, 23)(  9, 22)( 10, 28)( 11, 27)( 12, 26)( 13, 25)( 14, 24)( 15, 37)
( 16, 36)( 17, 42)( 18, 41)( 19, 40)( 20, 39)( 21, 38)( 43, 51)( 44, 50)
( 45, 56)( 46, 55)( 47, 54)( 48, 53)( 49, 52)( 57, 58)( 59, 63)( 60, 62)
( 64, 93)( 65, 92)( 66, 98)( 67, 97)( 68, 96)( 69, 95)( 70, 94)( 71, 86)
( 72, 85)( 73, 91)( 74, 90)( 75, 89)( 76, 88)( 77, 87)( 78,100)( 79, 99)
( 80,105)( 81,104)( 82,103)( 83,102)( 84,101)(106,114)(107,113)(108,119)
(109,118)(110,117)(111,116)(112,115)(120,121)(122,126)(123,125)(127,156)
(128,155)(129,161)(130,160)(131,159)(132,158)(133,157)(134,149)(135,148)
(136,154)(137,153)(138,152)(139,151)(140,150)(141,163)(142,162)(143,168)
(144,167)(145,166)(146,165)(147,164)(169,177)(170,176)(171,182)(172,181)
(173,180)(174,179)(175,178)(183,184)(185,189)(186,188);
s2 := 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)( 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)( 85,169)( 86,170)( 87,171)( 88,172)( 89,173)
( 90,174)( 91,175)( 92,176)( 93,177)( 94,178)( 95,179)( 96,180)( 97,181)
( 98,182)( 99,183)(100,184)(101,185)(102,186)(103,187)(104,188)(105,189)
(106,148)(107,149)(108,150)(109,151)(110,152)(111,153)(112,154)(113,155)
(114,156)(115,157)(116,158)(117,159)(118,160)(119,161)(120,162)(121,163)
(122,164)(123,165)(124,166)(125,167)(126,168);
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, 85)( 23, 86)
( 24, 87)( 25, 88)( 26, 89)( 27, 90)( 28, 91)( 29, 92)( 30, 93)( 31, 94)
( 32, 95)( 33, 96)( 34, 97)( 35, 98)( 36, 99)( 37,100)( 38,101)( 39,102)
( 40,103)( 41,104)( 42,105)( 43,106)( 44,107)( 45,108)( 46,109)( 47,110)
( 48,111)( 49,112)( 50,113)( 51,114)( 52,115)( 53,116)( 54,117)( 55,118)
( 56,119)( 57,120)( 58,121)( 59,122)( 60,123)( 61,124)( 62,125)( 63,126);
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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