Polytope of Type {2,9,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,6,6}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1858)
Rank : 5
Schlafli Type : {2,9,6,6}
Number of vertices, edges, etc : 2, 9, 27, 18, 6
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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) :
   2-fold quotients : {2,9,6,3}*648
   3-fold quotients : {2,9,2,6}*432, {2,3,6,6}*432a
   6-fold quotients : {2,9,2,3}*216, {2,3,6,3}*216
   9-fold quotients : {2,9,2,2}*144, {2,3,2,6}*144
   18-fold quotients : {2,3,2,3}*72
   27-fold quotients : {2,3,2,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 63)( 31, 65)( 32, 64)( 33, 60)
( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 72)( 40, 74)( 41, 73)
( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 81)( 49, 83)
( 50, 82)( 51, 78)( 52, 80)( 53, 79)( 54, 75)( 55, 77)( 56, 76)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,144)(112,146)(113,145)(114,141)(115,143)
(116,142)(117,138)(118,140)(119,139)(120,153)(121,155)(122,154)(123,150)
(124,152)(125,151)(126,147)(127,149)(128,148)(129,162)(130,164)(131,163)
(132,159)(133,161)(134,160)(135,156)(136,158)(137,157);;
s2 := (  3, 30)(  4, 32)(  5, 31)(  6, 36)(  7, 38)(  8, 37)(  9, 33)( 10, 35)
( 11, 34)( 12, 40)( 13, 39)( 14, 41)( 15, 46)( 16, 45)( 17, 47)( 18, 43)
( 19, 42)( 20, 44)( 21, 50)( 22, 49)( 23, 48)( 24, 56)( 25, 55)( 26, 54)
( 27, 53)( 28, 52)( 29, 51)( 57, 63)( 58, 65)( 59, 64)( 61, 62)( 66, 73)
( 67, 72)( 68, 74)( 69, 70)( 75, 83)( 76, 82)( 77, 81)( 78, 80)( 84,111)
( 85,113)( 86,112)( 87,117)( 88,119)( 89,118)( 90,114)( 91,116)( 92,115)
( 93,121)( 94,120)( 95,122)( 96,127)( 97,126)( 98,128)( 99,124)(100,123)
(101,125)(102,131)(103,130)(104,129)(105,137)(106,136)(107,135)(108,134)
(109,133)(110,132)(138,144)(139,146)(140,145)(142,143)(147,154)(148,153)
(149,155)(150,151)(156,164)(157,163)(158,162)(159,161);;
s3 := (  3, 12)(  4, 14)(  5, 13)(  6, 15)(  7, 17)(  8, 16)(  9, 18)( 10, 20)
( 11, 19)( 22, 23)( 25, 26)( 28, 29)( 30, 39)( 31, 41)( 32, 40)( 33, 42)
( 34, 44)( 35, 43)( 36, 45)( 37, 47)( 38, 46)( 49, 50)( 52, 53)( 55, 56)
( 57, 66)( 58, 68)( 59, 67)( 60, 69)( 61, 71)( 62, 70)( 63, 72)( 64, 74)
( 65, 73)( 76, 77)( 79, 80)( 82, 83)( 84, 93)( 85, 95)( 86, 94)( 87, 96)
( 88, 98)( 89, 97)( 90, 99)( 91,101)( 92,100)(103,104)(106,107)(109,110)
(111,120)(112,122)(113,121)(114,123)(115,125)(116,124)(117,126)(118,128)
(119,127)(130,131)(133,134)(136,137)(138,147)(139,149)(140,148)(141,150)
(142,152)(143,151)(144,153)(145,155)(146,154)(157,158)(160,161)(163,164);;
s4 := (  3, 84)(  4, 86)(  5, 85)(  6, 87)(  7, 89)(  8, 88)(  9, 90)( 10, 92)
( 11, 91)( 12,102)( 13,104)( 14,103)( 15,105)( 16,107)( 17,106)( 18,108)
( 19,110)( 20,109)( 21, 93)( 22, 95)( 23, 94)( 24, 96)( 25, 98)( 26, 97)
( 27, 99)( 28,101)( 29,100)( 30,111)( 31,113)( 32,112)( 33,114)( 34,116)
( 35,115)( 36,117)( 37,119)( 38,118)( 39,129)( 40,131)( 41,130)( 42,132)
( 43,134)( 44,133)( 45,135)( 46,137)( 47,136)( 48,120)( 49,122)( 50,121)
( 51,123)( 52,125)( 53,124)( 54,126)( 55,128)( 56,127)( 57,138)( 58,140)
( 59,139)( 60,141)( 61,143)( 62,142)( 63,144)( 64,146)( 65,145)( 66,156)
( 67,158)( 68,157)( 69,159)( 70,161)( 71,160)( 72,162)( 73,164)( 74,163)
( 75,147)( 76,149)( 77,148)( 78,150)( 79,152)( 80,151)( 81,153)( 82,155)
( 83,154);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
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(164)!(1,2);
s1 := Sym(164)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 63)( 31, 65)( 32, 64)
( 33, 60)( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 72)( 40, 74)
( 41, 73)( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 81)
( 49, 83)( 50, 82)( 51, 78)( 52, 80)( 53, 79)( 54, 75)( 55, 77)( 56, 76)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,144)(112,146)(113,145)(114,141)
(115,143)(116,142)(117,138)(118,140)(119,139)(120,153)(121,155)(122,154)
(123,150)(124,152)(125,151)(126,147)(127,149)(128,148)(129,162)(130,164)
(131,163)(132,159)(133,161)(134,160)(135,156)(136,158)(137,157);
s2 := Sym(164)!(  3, 30)(  4, 32)(  5, 31)(  6, 36)(  7, 38)(  8, 37)(  9, 33)
( 10, 35)( 11, 34)( 12, 40)( 13, 39)( 14, 41)( 15, 46)( 16, 45)( 17, 47)
( 18, 43)( 19, 42)( 20, 44)( 21, 50)( 22, 49)( 23, 48)( 24, 56)( 25, 55)
( 26, 54)( 27, 53)( 28, 52)( 29, 51)( 57, 63)( 58, 65)( 59, 64)( 61, 62)
( 66, 73)( 67, 72)( 68, 74)( 69, 70)( 75, 83)( 76, 82)( 77, 81)( 78, 80)
( 84,111)( 85,113)( 86,112)( 87,117)( 88,119)( 89,118)( 90,114)( 91,116)
( 92,115)( 93,121)( 94,120)( 95,122)( 96,127)( 97,126)( 98,128)( 99,124)
(100,123)(101,125)(102,131)(103,130)(104,129)(105,137)(106,136)(107,135)
(108,134)(109,133)(110,132)(138,144)(139,146)(140,145)(142,143)(147,154)
(148,153)(149,155)(150,151)(156,164)(157,163)(158,162)(159,161);
s3 := Sym(164)!(  3, 12)(  4, 14)(  5, 13)(  6, 15)(  7, 17)(  8, 16)(  9, 18)
( 10, 20)( 11, 19)( 22, 23)( 25, 26)( 28, 29)( 30, 39)( 31, 41)( 32, 40)
( 33, 42)( 34, 44)( 35, 43)( 36, 45)( 37, 47)( 38, 46)( 49, 50)( 52, 53)
( 55, 56)( 57, 66)( 58, 68)( 59, 67)( 60, 69)( 61, 71)( 62, 70)( 63, 72)
( 64, 74)( 65, 73)( 76, 77)( 79, 80)( 82, 83)( 84, 93)( 85, 95)( 86, 94)
( 87, 96)( 88, 98)( 89, 97)( 90, 99)( 91,101)( 92,100)(103,104)(106,107)
(109,110)(111,120)(112,122)(113,121)(114,123)(115,125)(116,124)(117,126)
(118,128)(119,127)(130,131)(133,134)(136,137)(138,147)(139,149)(140,148)
(141,150)(142,152)(143,151)(144,153)(145,155)(146,154)(157,158)(160,161)
(163,164);
s4 := Sym(164)!(  3, 84)(  4, 86)(  5, 85)(  6, 87)(  7, 89)(  8, 88)(  9, 90)
( 10, 92)( 11, 91)( 12,102)( 13,104)( 14,103)( 15,105)( 16,107)( 17,106)
( 18,108)( 19,110)( 20,109)( 21, 93)( 22, 95)( 23, 94)( 24, 96)( 25, 98)
( 26, 97)( 27, 99)( 28,101)( 29,100)( 30,111)( 31,113)( 32,112)( 33,114)
( 34,116)( 35,115)( 36,117)( 37,119)( 38,118)( 39,129)( 40,131)( 41,130)
( 42,132)( 43,134)( 44,133)( 45,135)( 46,137)( 47,136)( 48,120)( 49,122)
( 50,121)( 51,123)( 52,125)( 53,124)( 54,126)( 55,128)( 56,127)( 57,138)
( 58,140)( 59,139)( 60,141)( 61,143)( 62,142)( 63,144)( 64,146)( 65,145)
( 66,156)( 67,158)( 68,157)( 69,159)( 70,161)( 71,160)( 72,162)( 73,164)
( 74,163)( 75,147)( 76,149)( 77,148)( 78,150)( 79,152)( 80,151)( 81,153)
( 82,155)( 83,154);
poly := sub<Sym(164)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope