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Polytope of Type {9,6,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,14}*1512
if this polytope has a name.
Group : SmallGroup(1512,485)
Rank : 4
Schlafli Type : {9,6,14}
Number of vertices, edges, etc : 9, 27, 42, 14
Order of s0s1s2s3 : 126
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 : {9,2,14}*504, {3,6,14}*504
   6-fold quotients : {9,2,7}*252
   7-fold quotients : {9,6,2}*216
   9-fold quotients : {3,2,14}*168
   18-fold quotients : {3,2,7}*84
   21-fold quotients : {9,2,2}*72, {3,6,2}*72
   63-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 22, 43)
( 23, 45)( 24, 44)( 25, 46)( 26, 48)( 27, 47)( 28, 49)( 29, 51)( 30, 50)
( 31, 52)( 32, 54)( 33, 53)( 34, 55)( 35, 57)( 36, 56)( 37, 58)( 38, 60)
( 39, 59)( 40, 61)( 41, 63)( 42, 62)( 64,128)( 65,127)( 66,129)( 67,131)
( 68,130)( 69,132)( 70,134)( 71,133)( 72,135)( 73,137)( 74,136)( 75,138)
( 76,140)( 77,139)( 78,141)( 79,143)( 80,142)( 81,144)( 82,146)( 83,145)
( 84,147)( 85,170)( 86,169)( 87,171)( 88,173)( 89,172)( 90,174)( 91,176)
( 92,175)( 93,177)( 94,179)( 95,178)( 96,180)( 97,182)( 98,181)( 99,183)
(100,185)(101,184)(102,186)(103,188)(104,187)(105,189)(106,149)(107,148)
(108,150)(109,152)(110,151)(111,153)(112,155)(113,154)(114,156)(115,158)
(116,157)(117,159)(118,161)(119,160)(120,162)(121,164)(122,163)(123,165)
(124,167)(125,166)(126,168);;
s1 := (  1, 85)(  2, 87)(  3, 86)(  4, 88)(  5, 90)(  6, 89)(  7, 91)(  8, 93)
(  9, 92)( 10, 94)( 11, 96)( 12, 95)( 13, 97)( 14, 99)( 15, 98)( 16,100)
( 17,102)( 18,101)( 19,103)( 20,105)( 21,104)( 22, 64)( 23, 66)( 24, 65)
( 25, 67)( 26, 69)( 27, 68)( 28, 70)( 29, 72)( 30, 71)( 31, 73)( 32, 75)
( 33, 74)( 34, 76)( 35, 78)( 36, 77)( 37, 79)( 38, 81)( 39, 80)( 40, 82)
( 41, 84)( 42, 83)( 43,106)( 44,108)( 45,107)( 46,109)( 47,111)( 48,110)
( 49,112)( 50,114)( 51,113)( 52,115)( 53,117)( 54,116)( 55,118)( 56,120)
( 57,119)( 58,121)( 59,123)( 60,122)( 61,124)( 62,126)( 63,125)(127,149)
(128,148)(129,150)(130,152)(131,151)(132,153)(133,155)(134,154)(135,156)
(136,158)(137,157)(138,159)(139,161)(140,160)(141,162)(142,164)(143,163)
(144,165)(145,167)(146,166)(147,168)(169,170)(172,173)(175,176)(178,179)
(181,182)(184,185)(187,188);;
s2 := (  4, 19)(  5, 20)(  6, 21)(  7, 16)(  8, 17)(  9, 18)( 10, 13)( 11, 14)
( 12, 15)( 22, 43)( 23, 44)( 24, 45)( 25, 61)( 26, 62)( 27, 63)( 28, 58)
( 29, 59)( 30, 60)( 31, 55)( 32, 56)( 33, 57)( 34, 52)( 35, 53)( 36, 54)
( 37, 49)( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 67, 82)( 68, 83)
( 69, 84)( 70, 79)( 71, 80)( 72, 81)( 73, 76)( 74, 77)( 75, 78)( 85,106)
( 86,107)( 87,108)( 88,124)( 89,125)( 90,126)( 91,121)( 92,122)( 93,123)
( 94,118)( 95,119)( 96,120)( 97,115)( 98,116)( 99,117)(100,112)(101,113)
(102,114)(103,109)(104,110)(105,111)(130,145)(131,146)(132,147)(133,142)
(134,143)(135,144)(136,139)(137,140)(138,141)(148,169)(149,170)(150,171)
(151,187)(152,188)(153,189)(154,184)(155,185)(156,186)(157,181)(158,182)
(159,183)(160,178)(161,179)(162,180)(163,175)(164,176)(165,177)(166,172)
(167,173)(168,174);;
s3 := (  1,  4)(  2,  5)(  3,  6)(  7, 19)(  8, 20)(  9, 21)( 10, 16)( 11, 17)
( 12, 18)( 22, 25)( 23, 26)( 24, 27)( 28, 40)( 29, 41)( 30, 42)( 31, 37)
( 32, 38)( 33, 39)( 43, 46)( 44, 47)( 45, 48)( 49, 61)( 50, 62)( 51, 63)
( 52, 58)( 53, 59)( 54, 60)( 64, 67)( 65, 68)( 66, 69)( 70, 82)( 71, 83)
( 72, 84)( 73, 79)( 74, 80)( 75, 81)( 85, 88)( 86, 89)( 87, 90)( 91,103)
( 92,104)( 93,105)( 94,100)( 95,101)( 96,102)(106,109)(107,110)(108,111)
(112,124)(113,125)(114,126)(115,121)(116,122)(117,123)(127,130)(128,131)
(129,132)(133,145)(134,146)(135,147)(136,142)(137,143)(138,144)(148,151)
(149,152)(150,153)(154,166)(155,167)(156,168)(157,163)(158,164)(159,165)
(169,172)(170,173)(171,174)(175,187)(176,188)(177,189)(178,184)(179,185)
(180,186);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(189)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 22, 43)( 23, 45)( 24, 44)( 25, 46)( 26, 48)( 27, 47)( 28, 49)( 29, 51)
( 30, 50)( 31, 52)( 32, 54)( 33, 53)( 34, 55)( 35, 57)( 36, 56)( 37, 58)
( 38, 60)( 39, 59)( 40, 61)( 41, 63)( 42, 62)( 64,128)( 65,127)( 66,129)
( 67,131)( 68,130)( 69,132)( 70,134)( 71,133)( 72,135)( 73,137)( 74,136)
( 75,138)( 76,140)( 77,139)( 78,141)( 79,143)( 80,142)( 81,144)( 82,146)
( 83,145)( 84,147)( 85,170)( 86,169)( 87,171)( 88,173)( 89,172)( 90,174)
( 91,176)( 92,175)( 93,177)( 94,179)( 95,178)( 96,180)( 97,182)( 98,181)
( 99,183)(100,185)(101,184)(102,186)(103,188)(104,187)(105,189)(106,149)
(107,148)(108,150)(109,152)(110,151)(111,153)(112,155)(113,154)(114,156)
(115,158)(116,157)(117,159)(118,161)(119,160)(120,162)(121,164)(122,163)
(123,165)(124,167)(125,166)(126,168);
s1 := Sym(189)!(  1, 85)(  2, 87)(  3, 86)(  4, 88)(  5, 90)(  6, 89)(  7, 91)
(  8, 93)(  9, 92)( 10, 94)( 11, 96)( 12, 95)( 13, 97)( 14, 99)( 15, 98)
( 16,100)( 17,102)( 18,101)( 19,103)( 20,105)( 21,104)( 22, 64)( 23, 66)
( 24, 65)( 25, 67)( 26, 69)( 27, 68)( 28, 70)( 29, 72)( 30, 71)( 31, 73)
( 32, 75)( 33, 74)( 34, 76)( 35, 78)( 36, 77)( 37, 79)( 38, 81)( 39, 80)
( 40, 82)( 41, 84)( 42, 83)( 43,106)( 44,108)( 45,107)( 46,109)( 47,111)
( 48,110)( 49,112)( 50,114)( 51,113)( 52,115)( 53,117)( 54,116)( 55,118)
( 56,120)( 57,119)( 58,121)( 59,123)( 60,122)( 61,124)( 62,126)( 63,125)
(127,149)(128,148)(129,150)(130,152)(131,151)(132,153)(133,155)(134,154)
(135,156)(136,158)(137,157)(138,159)(139,161)(140,160)(141,162)(142,164)
(143,163)(144,165)(145,167)(146,166)(147,168)(169,170)(172,173)(175,176)
(178,179)(181,182)(184,185)(187,188);
s2 := Sym(189)!(  4, 19)(  5, 20)(  6, 21)(  7, 16)(  8, 17)(  9, 18)( 10, 13)
( 11, 14)( 12, 15)( 22, 43)( 23, 44)( 24, 45)( 25, 61)( 26, 62)( 27, 63)
( 28, 58)( 29, 59)( 30, 60)( 31, 55)( 32, 56)( 33, 57)( 34, 52)( 35, 53)
( 36, 54)( 37, 49)( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 67, 82)
( 68, 83)( 69, 84)( 70, 79)( 71, 80)( 72, 81)( 73, 76)( 74, 77)( 75, 78)
( 85,106)( 86,107)( 87,108)( 88,124)( 89,125)( 90,126)( 91,121)( 92,122)
( 93,123)( 94,118)( 95,119)( 96,120)( 97,115)( 98,116)( 99,117)(100,112)
(101,113)(102,114)(103,109)(104,110)(105,111)(130,145)(131,146)(132,147)
(133,142)(134,143)(135,144)(136,139)(137,140)(138,141)(148,169)(149,170)
(150,171)(151,187)(152,188)(153,189)(154,184)(155,185)(156,186)(157,181)
(158,182)(159,183)(160,178)(161,179)(162,180)(163,175)(164,176)(165,177)
(166,172)(167,173)(168,174);
s3 := Sym(189)!(  1,  4)(  2,  5)(  3,  6)(  7, 19)(  8, 20)(  9, 21)( 10, 16)
( 11, 17)( 12, 18)( 22, 25)( 23, 26)( 24, 27)( 28, 40)( 29, 41)( 30, 42)
( 31, 37)( 32, 38)( 33, 39)( 43, 46)( 44, 47)( 45, 48)( 49, 61)( 50, 62)
( 51, 63)( 52, 58)( 53, 59)( 54, 60)( 64, 67)( 65, 68)( 66, 69)( 70, 82)
( 71, 83)( 72, 84)( 73, 79)( 74, 80)( 75, 81)( 85, 88)( 86, 89)( 87, 90)
( 91,103)( 92,104)( 93,105)( 94,100)( 95,101)( 96,102)(106,109)(107,110)
(108,111)(112,124)(113,125)(114,126)(115,121)(116,122)(117,123)(127,130)
(128,131)(129,132)(133,145)(134,146)(135,147)(136,142)(137,143)(138,144)
(148,151)(149,152)(150,153)(154,166)(155,167)(156,168)(157,163)(158,164)
(159,165)(169,172)(170,173)(171,174)(175,187)(176,188)(177,189)(178,184)
(179,185)(180,186);
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
to this polytope