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Polytope of Type {9,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,6}*648a
if this polytope has a name.
Group : SmallGroup(648,297)
Rank : 4
Schlafli Type : {9,6,6}
Number of vertices, edges, etc : 9, 27, 18, 6
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 6
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {9,6,6,2} of size 1296
Vertex Figure Of :
   {2,9,6,6} of size 1296
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {9,6,3}*324
   3-fold quotients : {9,2,6}*216, {3,6,6}*216a
   6-fold quotients : {9,2,3}*108, {3,6,3}*108
   9-fold quotients : {9,2,2}*72, {3,2,6}*72
   18-fold quotients : {3,2,3}*36
   27-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {9,6,12}*1296a, {18,6,6}*1296a
   3-fold covers : {9,6,18}*1944a, {27,6,6}*1944a, {9,6,6}*1944a, {9,6,6}*1944b, {9,6,6}*1944d
Permutation Representation (GAP) :
s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 61)( 29, 63)( 30, 62)( 31, 58)
( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 70)( 38, 72)( 39, 71)
( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 46, 79)( 47, 81)
( 48, 80)( 49, 76)( 50, 78)( 51, 77)( 52, 73)( 53, 75)( 54, 74)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,142)(110,144)(111,143)(112,139)(113,141)
(114,140)(115,136)(116,138)(117,137)(118,151)(119,153)(120,152)(121,148)
(122,150)(123,149)(124,145)(125,147)(126,146)(127,160)(128,162)(129,161)
(130,157)(131,159)(132,158)(133,154)(134,156)(135,155);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)(  8, 33)
(  9, 32)( 10, 38)( 11, 37)( 12, 39)( 13, 44)( 14, 43)( 15, 45)( 16, 41)
( 17, 40)( 18, 42)( 19, 48)( 20, 47)( 21, 46)( 22, 54)( 23, 53)( 24, 52)
( 25, 51)( 26, 50)( 27, 49)( 55, 61)( 56, 63)( 57, 62)( 59, 60)( 64, 71)
( 65, 70)( 66, 72)( 67, 68)( 73, 81)( 74, 80)( 75, 79)( 76, 78)( 82,109)
( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)( 90,113)
( 91,119)( 92,118)( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)( 98,121)
( 99,123)(100,129)(101,128)(102,127)(103,135)(104,134)(105,133)(106,132)
(107,131)(108,130)(136,142)(137,144)(138,143)(140,141)(145,152)(146,151)
(147,153)(148,149)(154,162)(155,161)(156,160)(157,159);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)(  8, 18)
(  9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 40)
( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 47, 48)( 50, 51)( 53, 54)
( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)( 62, 72)
( 63, 71)( 74, 75)( 77, 78)( 80, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)
( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)(107,108)
(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)(116,126)
(117,125)(128,129)(131,132)(134,135)(136,145)(137,147)(138,146)(139,148)
(140,150)(141,149)(142,151)(143,153)(144,152)(155,156)(158,159)(161,162);;
s3 := (  1, 82)(  2, 84)(  3, 83)(  4, 85)(  5, 87)(  6, 86)(  7, 88)(  8, 90)
(  9, 89)( 10,100)( 11,102)( 12,101)( 13,103)( 14,105)( 15,104)( 16,106)
( 17,108)( 18,107)( 19, 91)( 20, 93)( 21, 92)( 22, 94)( 23, 96)( 24, 95)
( 25, 97)( 26, 99)( 27, 98)( 28,109)( 29,111)( 30,110)( 31,112)( 32,114)
( 33,113)( 34,115)( 35,117)( 36,116)( 37,127)( 38,129)( 39,128)( 40,130)
( 41,132)( 42,131)( 43,133)( 44,135)( 45,134)( 46,118)( 47,120)( 48,119)
( 49,121)( 50,123)( 51,122)( 52,124)( 53,126)( 54,125)( 55,136)( 56,138)
( 57,137)( 58,139)( 59,141)( 60,140)( 61,142)( 62,144)( 63,143)( 64,154)
( 65,156)( 66,155)( 67,157)( 68,159)( 69,158)( 70,160)( 71,162)( 72,161)
( 73,145)( 74,147)( 75,146)( 76,148)( 77,150)( 78,149)( 79,151)( 80,153)
( 81,152);;
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*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(162)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 61)( 29, 63)( 30, 62)
( 31, 58)( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 70)( 38, 72)
( 39, 71)( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 46, 79)
( 47, 81)( 48, 80)( 49, 76)( 50, 78)( 51, 77)( 52, 73)( 53, 75)( 54, 74)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,142)(110,144)(111,143)(112,139)
(113,141)(114,140)(115,136)(116,138)(117,137)(118,151)(119,153)(120,152)
(121,148)(122,150)(123,149)(124,145)(125,147)(126,146)(127,160)(128,162)
(129,161)(130,157)(131,159)(132,158)(133,154)(134,156)(135,155);
s1 := Sym(162)!(  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)
(  8, 33)(  9, 32)( 10, 38)( 11, 37)( 12, 39)( 13, 44)( 14, 43)( 15, 45)
( 16, 41)( 17, 40)( 18, 42)( 19, 48)( 20, 47)( 21, 46)( 22, 54)( 23, 53)
( 24, 52)( 25, 51)( 26, 50)( 27, 49)( 55, 61)( 56, 63)( 57, 62)( 59, 60)
( 64, 71)( 65, 70)( 66, 72)( 67, 68)( 73, 81)( 74, 80)( 75, 79)( 76, 78)
( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)
( 90,113)( 91,119)( 92,118)( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)
( 98,121)( 99,123)(100,129)(101,128)(102,127)(103,135)(104,134)(105,133)
(106,132)(107,131)(108,130)(136,142)(137,144)(138,143)(140,141)(145,152)
(146,151)(147,153)(148,149)(154,162)(155,161)(156,160)(157,159);
s2 := Sym(162)!(  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)
(  8, 18)(  9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 40)( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 47, 48)( 50, 51)
( 53, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)
( 62, 72)( 63, 71)( 74, 75)( 77, 78)( 80, 81)( 82, 91)( 83, 93)( 84, 92)
( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(101,102)(104,105)
(107,108)(109,118)(110,120)(111,119)(112,121)(113,123)(114,122)(115,124)
(116,126)(117,125)(128,129)(131,132)(134,135)(136,145)(137,147)(138,146)
(139,148)(140,150)(141,149)(142,151)(143,153)(144,152)(155,156)(158,159)
(161,162);
s3 := Sym(162)!(  1, 82)(  2, 84)(  3, 83)(  4, 85)(  5, 87)(  6, 86)(  7, 88)
(  8, 90)(  9, 89)( 10,100)( 11,102)( 12,101)( 13,103)( 14,105)( 15,104)
( 16,106)( 17,108)( 18,107)( 19, 91)( 20, 93)( 21, 92)( 22, 94)( 23, 96)
( 24, 95)( 25, 97)( 26, 99)( 27, 98)( 28,109)( 29,111)( 30,110)( 31,112)
( 32,114)( 33,113)( 34,115)( 35,117)( 36,116)( 37,127)( 38,129)( 39,128)
( 40,130)( 41,132)( 42,131)( 43,133)( 44,135)( 45,134)( 46,118)( 47,120)
( 48,119)( 49,121)( 50,123)( 51,122)( 52,124)( 53,126)( 54,125)( 55,136)
( 56,138)( 57,137)( 58,139)( 59,141)( 60,140)( 61,142)( 62,144)( 63,143)
( 64,154)( 65,156)( 66,155)( 67,157)( 68,159)( 69,158)( 70,160)( 71,162)
( 72,161)( 73,145)( 74,147)( 75,146)( 76,148)( 77,150)( 78,149)( 79,151)
( 80,153)( 81,152);
poly := sub<Sym(162)|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*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
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 >; 
 
References : None.
to this polytope