Polytope of Type {6,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,3}*648a
if this polytope has a name.
Group : SmallGroup(648,299)
Rank : 4
Schlafli Type : {6,6,3}
Number of vertices, edges, etc : 18, 54, 27, 3
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,3,2} of size 1296
Vertex Figure Of :
   {2,6,6,3} of size 1296
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,3}*324b
   3-fold quotients : {6,6,3}*216a
   6-fold quotients : {3,6,3}*108
   9-fold quotients : {6,2,3}*72
   18-fold quotients : {3,2,3}*36
   27-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,6,3}*1296a, {6,6,6}*1296a
   3-fold covers : {6,6,3}*1944a, {18,6,3}*1944a, {6,6,9}*1944b, {6,6,3}*1944e
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope