6 Interfaces to Other Data Formats for Character Tables

This chapter describes data formats for character tables that can be read or created by GAP. Currently these are the formats used by

• the CAS system (see 6.1),

• the MOC system (see 6.2),

• GAP 3 (see 6.3),

• the so-called Cambridge format (see 6.4), and

• the MAGMA system (see 6.5).

6.1 Interface to the CAS System

The interface to CAS (see [NPP84]) is thought just for printing the CAS data to a file. The function CASString (6.1-1) is available mainly in order to document the data format. Reading CAS tables is not supported; note that the tables contained in the CAS Character Table Library have been migrated to GAP using a few sed scripts and C programs.

6.1-1 CASString

is a string that encodes the CAS library format of the character table tbl. This string can be printed to a file which then can be read into the CAS system using its get command (see [NPP84]).

The used line length is the first entry in the list returned by SizeScreen (Reference: SizeScreen).

Only the known values of the following attributes are used. ClassParameters (Reference: ClassParameters) (for partitions only), ComputedClassFusions (Reference: ComputedClassFusions), ComputedIndicators (Reference: ComputedIndicators), ComputedPowerMaps (Reference: ComputedPowerMaps), ComputedPrimeBlocks (Reference: ComputedPrimeBlockss), Identifier (Reference: Identifier (for character tables)), InfoText (Reference: InfoText), Irr (Reference: Irr), OrdersClassRepresentatives (Reference: OrdersClassRepresentatives), Size (Reference: Size), SizesCentralizers (Reference: SizesCentralizers).

gap> Print( CASString( CharacterTable( "Cyclic", 2 ) ), "\n" );
'C2'
00/00/00. 00.00.00.
(2,2,0,2,-1,0)
text:
(#computed using generic character table for cyclic groups#),
order=2,
centralizers:(
2,2
),
reps:(
1,2
),
powermap:2(
1,1
),
characters:
(1,1
,0:0)
(1,-1
,0:0);
/// converted from GAP

6.2 Interface to the MOC System

The interface to MOC (see [HJLP]) can be used to print MOC input. Additionally it provides an alternative representation of (virtual) characters.

The MOC 3 code of a 5 digit number in MOC 2 code is given by the following list. (Note that the code must contain only lower case letters.)

ABCD    for  0ABCD
a       for  10000
b       for  10001          k       for  20001
c       for  10002          l       for  20002
d       for  10003          m       for  20003
e       for  10004          n       for  20004
f       for  10005          o       for  20005
g       for  10006          p       for  20006
h       for  10007          q       for  20007
i       for  10008          r       for  20008
j       for  10009          s       for  20009
tAB     for  100AB
uAB     for  200AB
vABCD   for  1ABCD
wABCD   for  2ABCD
yABC    for  30ABC
z       for  31000

Note that any long number in MOC 2 format is divided into packages of length 4, the first (!) one filled with leading zeros if necessary. Such a number with decimals d_1, d_2, ..., d_{4n+k} is the sequence 0 d_1 d_2 d_3 d_4 ... 0 d_{4n-3} d_{4n-2} d_{4n-1} d_4n d_{4n+1} ... d_{4n+k} where 0 ≤ k ≤ 3, the first digit of x is 1 if the number is positive and 2 if the number is negative, and then follow (4-k) zeros.

Details about the MOC system are explained in [HJLP], a brief description can be found in [LP91].

6.2-1 MAKElb11

For a list listofns of positive integers, MAKElb11 prints field information for all number fields with conductor in this list.

The output of MAKElb11 is used by the MOC system; Calling MAKElb11( [ 3 .. 189 ] ) will print something very similar to Richard Parker's file lb11.

gap> MAKElb11( [ 3, 4 ] );
   3   2   0   1   0
   4   2   0   1   0

6.2-2 MOCTable

MOCTable returns the MOC table record of the GAP character table gaptbl.

The one argument version can be used only if gaptbl is an ordinary (G.0) table. For Brauer (G.p) tables, one has to specify a basic set basicset of ordinary irreducibles. basicset must then be a list of positions of the basic set characters in the Irr (Reference: Irr) list of the ordinary table of gaptbl.

The result is a record that contains the information of gaptbl in a format similar to the MOC 3 format. This record can, e. g., easily be printed out or be used to print out characters using MOCString (6.2-3).

The components of the result are

identifier

the string MOCTable( name ) where name is the Identifier (Reference: Identifier (for character tables)) value of gaptbl,

GAPtbl

gaptbl,

prime

the characteristic of the field (label 30105 in MOC),

centralizers

centralizer orders for cyclic subgroups (label 30130)

orders

element orders for cyclic subgroups (label 30140)

fieldbases

at position i the Parker basis of the number field generated by the character values of the i-th cyclic subgroup. The length of fieldbases is equal to the value of label 30110 in MOC.

cycsubgps

cycsubgps[i] = j means that class i of the GAP table belongs to the j-th cyclic subgroup of the GAP table,

repcycsub

repcycsub[j] = i means that class i of the GAP table is the representative of the j-th cyclic subgroup of the GAP table. Note that the representatives of GAP table and MOC table need not agree!

galconjinfo

a list [ r_1, c_1, r_2, c_2, ..., r_n, c_n ] which means that the i-th class of the GAP table is the c_i-th conjugate of the representative of the r_i-th cyclic subgroup on the MOC table. (This is used to translate back to GAP format, stored under label 30160)

30170

(power maps) for each cyclic subgroup (except the trivial one) and each prime divisor of the representative order store four values, namely the number of the subgroup, the power, the number of the cyclic subgroup containing the image, and the power to which the representative must be raised to yield the image class. (This is used only to construct the 30230 power map/embedding information.) In 30170 only a list of lists (one for each cyclic subgroup) of all these values is stored, it will not be used by GAP.

tensinfo

tensor product information, used to compute the coefficients of the Parker base for tensor products of characters (label 30210 in MOC). For a field with vector space basis (v_1, v_2, ..., v_n), the tensor product information of a cyclic subgroup in MOC (as computed by fct) is either 1 (for rational classes) or a sequence

n x_1,1 y_1,1 z_1,1 x_1,2 y_1,2 z_1,2 ... x_1,m_1 y_1,m_1 z_1,m_1 0 x_2,1 y_2,1 z_2,1 x_2,2 y_2,2 z_2,2 ... x_2,m_2 y_2,m_2 z_2,m_2 0 ... z_n,m_n 0

which means that the coefficient of v_k in the product

( ∑_i=1^n a_i v_i ) ( ∑_j=1^n b_j v_j )

is equal to

∑_i=1^m_k x_k,i a_y_k,i} b_z_k,i} .

On a MOC table in GAP, the tensinfo component is a list of lists, each containing exactly the sequence mentioned above.

invmap

inverse map to compute complex conjugate characters, label 30220 in MOC.

powerinfo

field embeddings for p-th symmetrizations, p a prime integer not larger than the largest element order, label 30230 in MOC.

30900

basic set of restricted ordinary irreducibles in the case of nonzero characteristic, all ordinary irreducibles otherwise.

6.2-3 MOCString

Let moctbl be a MOC table record, as returned by MOCTable (6.2-2). MOCString returns a string describing the MOC 3 format of moctbl.

If a second argument chars is specified, it must be a list of MOC format characters as returned by MOCChars (6.2-6). In this case, these characters are stored under label 30900. If the second argument is missing then the basic set of ordinary irreducibles is stored under this label.

gap> moca5:= MOCTable( CharacterTable( "A5" ) );
rec( 30170 := [ [  ], [ 2, 2, 1, 1 ], [ 3, 3, 1, 1 ], [ 4, 5, 1, 1 ] ]
    , 
  30900 := [ [ 1, 1, 1, 1, 0 ], [ 3, -1, 0, 0, -1 ], 
      [ 3, -1, 0, 1, 1 ], [ 4, 0, 1, -1, 0 ], [ 5, 1, -1, 0, 0 ] ], 
  GAPtbl := CharacterTable( "A5" ), centralizers := [ 60, 4, 3, 5 ], 
  cycsubgps := [ 1, 2, 3, 4, 4 ], 
  fieldbases := 
    [ CanonicalBasis( Rationals ), CanonicalBasis( Rationals ), 
      CanonicalBasis( Rationals ), 
      Basis( NF(5,[ 1, 4 ]), [ 1, E(5)+E(5)^4 ] ) ], fields := [  ], 
  galconjinfo := [ 1, 1, 2, 1, 3, 1, 4, 1, 4, 2 ], 
  identifier := "MOCTable(A5)", 
  invmap := [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], 
      [ 1, 4, 0, 1, 5, 0 ] ], orders := [ 1, 2, 3, 5 ], 
  powerinfo := 
    [ , 
      [ [ 1, 1, 0 ], [ 1, 1, 0 ], [ 1, 3, 0 ], 
          [ 1, 4, -1, 5, 0, -1, 5, 0 ] ], 
      [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 1, 0 ], 
          [ 1, 4, -1, 5, 0, -1, 5, 0 ] ],, 
      [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [ 1, 1, 0, 0 ] ] ], 
  prime := 0, repcycsub := [ 1, 2, 3, 4 ], 
  tensinfo := 
    [ [ 1 ], [ 1 ], [ 1 ], 
      [ 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, -1, 2, 2, 0 ] ] )
gap> str:= MOCString( moca5 );;
gap> str{[1..68]};
"y100y105ay110fey130t60edfy140bcdfy150bbbfcabbey160bbcbdbebecy170ccbb"
gap> moca5mod3:= MOCTable( CharacterTable( "A5" ) mod 3, [ 1 .. 4 ] );;
gap> MOCString( moca5mod3 ){ [ 1 .. 68 ] };
"y100y105dy110edy130t60efy140bcfy150bbfcabbey160bbcbdbdcy170ccbbdfbby"

6.2-4 ScanMOC

returns a record containing the information encoded in the list list. The components of the result are the labels that occur in list. If list is in MOC 2 format (10000-format), the names of components are 30000-numbers; if it is in MOC 3 format the names of components have yABC-format.

6.2-5 GAPChars

Let tbl be a character table or a MOC table record, and mocchars be either a list of MOC format characters (as returned by MOCChars (6.2-6)) or a list of positive integers such as a record component encoding characters, in a record produced by ScanMOC (6.2-4).

GAPChars returns translations of mocchars to GAP character values lists.

6.2-6 MOCChars

Let tbl be a character table or a MOC table record, and gapchars be a list of (GAP format) characters. MOCChars returns translations of gapchars to MOC format.

gap> scan:= ScanMOC( str );
rec( y050 := [ 5, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 1, 0, 0 ], 
  y105 := [ 0 ], y110 := [ 5, 4 ], y130 := [ 60, 4, 3, 5 ], 
  y140 := [ 1, 2, 3, 5 ], y150 := [ 1, 1, 1, 5, 2, 0, 1, 1, 4 ], 
  y160 := [ 1, 1, 2, 1, 3, 1, 4, 1, 4, 2 ], 
  y170 := [ 2, 2, 1, 1, 3, 3, 1, 1, 4, 5, 1, 1 ], 
  y210 := [ 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, -1, 2, 
      2, 0 ], y220 := [ 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0 ],
  y230 := [ 2, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 4, -1, 5, 0, -1, 5, 0 ], 
  y900 := [ 1, 1, 1, 1, 0, 3, -1, 0, 0, -1, 3, -1, 0, 1, 1, 4, 0, 1, 
      -1, 0, 5, 1, -1, 0, 0 ] )
gap> gapchars:= GAPChars( moca5, scan.y900 );
[ [ 1, 1, 1, 1, 1 ], [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ], 
  [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ], [ 4, 0, 1, -1, -1 ], 
  [ 5, 1, -1, 0, 0 ] ]
gap> mocchars:= MOCChars( moca5, gapchars );
[ [ 1, 1, 1, 1, 0 ], [ 3, -1, 0, 0, -1 ], [ 3, -1, 0, 1, 1 ], 
  [ 4, 0, 1, -1, 0 ], [ 5, 1, -1, 0, 0 ] ]
gap> Concatenation( mocchars ) = scan.y900;
true

6.3 Interface to GAP 3

The following functions are used to read and write character tables in GAP 3 format.

6.3-1 GAP3CharacterTableScan

Let string be a string that contains the output of the GAP 3 function PrintCharTable. In other words, string describes a GAP record whose components define an ordinary character table object in GAP 3. GAP3CharacterTableScan returns the corresponding GAP 4 character table object.

The supported record components are given by the list GAP3CharacterTableData (6.3-3).

6.3-2 GAP3CharacterTableString

For an ordinary character table tbl, GAP3CharacterTableString returns a string that when read into GAP 3 evaluates to a character table corresponding to tbl. A similar format is printed by the GAP 3 function PrintCharTable.

The supported record components are given by the list GAP3CharacterTableData (6.3-3).

gap> tbl:= CharacterTable( "Alternating", 5 );;
gap> str:= GAP3CharacterTableString( tbl );;
gap> Print( str );
rec(
centralizers := [ 60, 4, 3, 5, 5 ],
fusions := [ rec( map := [ 1, 3, 4, 7, 7 ], name := "Sym(5)" ) ],
identifier := "Alt(5)",
irreducibles := [
[ 1, 1, 1, 1, 1 ],
[ 4, 0, 1, -1, -1 ],
[ 5, 1, -1, 0, 0 ],
[ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ],
[ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ]
],
orders := [ 1, 2, 3, 5, 5 ],
powermap := [ , [ 1, 1, 3, 5, 4 ], [ 1, 2, 1, 5, 4 ], , [ 1, 2, 3, 1, \
1 ] ],
size := 60,
text := "computed using generic character table for alternating groups\
",
operations := CharTableOps )
gap> scan:= GAP3CharacterTableScan( str );
CharacterTable( "Alt(5)" )
gap> TransformingPermutationsCharacterTables( tbl, scan );
rec( columns := (), group := Group([ (4,5) ]), rows := () )

6.3-3 GAP3CharacterTableData

This is a list of pairs, the first entry being the name of a component in a GAP 3 character table and the second entry being the corresponding attribute name in GAP 4. The variable is used by GAP3CharacterTableScan (6.3-1) and GAP3CharacterTableString (6.3-2).

6.4 Interface to the Cambridge Format

The following functions deal with the so-called Cambridge format, in which the source data of the character tables in the Atlas of Finite Groups [CCNPW85] and in the Atlas of Brauer Characters [JLPW95] are stored. Each such table is stored on a file of its own. The line length is at most 78, and each item of the table starts in a new line, behind one of the following prefixes.

#23

a description and the name(s) of the simple group

#7

integers describing the column widths

#9

the symbols ; and @, denoting columns between tables and columns that belong to conjugacy classes, respectively

#1

the symbol | in columns between tables, and centralizer orders otherwise

#2

the symbols p (in the first column only), power (in the second column only, which belongs to the class of the identity element), | in other columns between tables, and descriptions of the powers of classes otherwise

#3

the symbols p' (in the first column only), part (in the second column only, which belongs to the class of the identity element), | in other columns between tables, and descriptions of the p-prime parts of classes otherwise

#4

the symbols ind and fus in columns between tables, and class names otherwise

#5

either | or strings composed from the symbols +, -, o, and integers in columns where the lines starting with #4 contain ind; the symbols :, ., ? in columns where these lines contain fus; character values or | otherwise

#6

the symbols |, ind, and, and fus in columns between tables; the symbol | and element orders of preimage classes in downward extensions otherwise

#8

the last line of the data, may contain the date of the last change

#C

comments.

6.4-1 CambridgeMaps

For a character table tbl, CambridgeMaps returns a record with the following components.

name

a list of strings denoting class names,

power

a list of strings, the i-th entry encodes the p-th powers of the i-th class, for all prime divisors p of the group order,

prime

a list of strings, the i-th entry encodes the p-prime parts of the i-th class, for all prime divisors p of the group order.

The meaning of the entries of the lists is defined in [CCNPW85, Chapter 7, Sections 3–5]).

CambridgeMaps is used for example by Display (Reference: Display (for a character table)) in the case that the powermap option has the value "ATLAS".

gap> CambridgeMaps( CharacterTable( "A5" ) );
rec( names := [ "1A", "2A", "3A", "5A", "B*" ], 
  power := [ "", "A", "A", "A", "A" ], 
  prime := [ "", "A", "A", "A", "A" ] )
gap> CambridgeMaps( CharacterTable( "A5" ) mod 2 );
rec( names := [ "1A", "3A", "5A", "B*" ], 
  power := [ "", "A", "A", "A" ], prime := [ "", "A", "A", "A" ] )

6.4-2 StringOfCambridgeFormat

(This function is experimental.)

Let tbls be a list of character tables, which are central extensions of the first entry in tbls, and such that the factor fusion to the first entry is stored on all other tables in the list.

StringOfCambridgeFormat returns a string that encodes an approximation of the Cambridge format file for the first entry in tbls. Differences to the original format may occur for irrational character values; the descriptions of these values have been chosen deliberately for the original files, it is not obvious how to compute these descriptions from the character tables in question.

gap> t:= CharacterTable( "A5" );;  2t:= CharacterTable( "2.A5" );;
gap> Print( StringOfCambridgeFormat( [ t, 2t ] ) );
#23 ? A5
#7 4 4 4 4 4 4 
#9 ; @ @ @ @ @ 
#1 | 60 4 3 5 5 
#2 p power A A A A 
#3 p' part A A A A 
#4 ind 1A 2A 3A 5A B* 
#5 + 1 1 1 1 1 
#5 + 3 -1 0 -b5 * 
#5 + 3 -1 0 * -b5 
#5 + 4 0 1 -1 -1 
#5 + 5 1 -1 0 0 
#6 ind 1 4 3 5 5 
#6 | 2 | 6 10 10 
#5 - 2 0 -1 b5 * 
#5 - 2 0 -1 * b5 
#5 - 4 0 1 -1 -1 
#5 - 6 0 0 1 1 
#8

6.5 Interface to the MAGMA System

This interface is intended to convert character tables given in MAGMA's display format into GAP character tables.

The function BosmaBase (6.5-1) is used for the translation of irrational values; this function may be of interest independent of the conversion of character tables.

6.5-1 BosmaBase

For a positive integer n that is not congruent to 2 modulo 4, BosmaBase returns the list of exponents i for which E(n)^i belongs to the canonical basis of the n-th cyclotomic field that is defined in [Bos90, Section 5].

As a set, this basis is defined as follows. Let P denote the set of prime divisors of n and n = ∏_{p ∈ P} n_p. Let e_l = E(l) for any positive integer l, and { e_{m_1}^j }_{j ∈ J} ⊗ { e_{m_2}^k }_{k ∈ K} = { e_{m_1}^j ⋅ e_{m_2}^k }_{j ∈ J, k ∈ K} for any positive integers m_1, m_2. (This notation is the same as the one used in the description of ZumbroichBase (Reference: ZumbroichBase).)

Then the basis is

B_n = ⨂_{p ∈ P} B_{n_p}

where

B_{n_p} = { e_{n_p}^k; 0 ≤ k ≤ φ(n_p)-1 };

here φ denotes Euler's function, see Phi (Reference: Phi).

B_n consists of roots of unity, it is an integral basis (that is, exactly the integral elements in ℚ_n have integral coefficients w.r.t. B_n, cf. IsIntegralCyclotomic (Reference: IsIntegralCyclotomic)), and for any divisor m of n that is not congruent to 2 modulo 4, B_m is a subset of B_n.

Note that the list l, say, that is returned by BosmaBase is in general not a set. The ordering of the elements in l fits to the coefficient lists for irrational values used by MAGMA's display format.

gap> b:= BosmaBase( 8 );
[ 0, 1, 2, 3 ]
gap> b:= Basis( CF(8), List( b, i -> E(8)^i ) );
Basis( CF(8), [ 1, E(8), E(4), E(8)^3 ] )
gap> Coefficients( b, Sqrt(2) );
[ 0, 1, 0, -1 ]
gap> Coefficients( b, Sqrt(-2) );
[ 0, 1, 0, 1 ]
gap> b:= BosmaBase( 15 );
[ 0, 5, 3, 8, 6, 11, 9, 14 ]
gap> b:= List( b, i -> E(15)^i );
[ 1, E(3), E(5), E(15)^8, E(5)^2, E(15)^11, E(5)^3, E(15)^14 ]
gap> Coefficients( Basis( CF(15), b ), EB(15) );
[ -1, -1, 0, 0, -1, -2, -1, -2 ]
gap> BosmaBase( 48 );
[ 0, 3, 6, 9, 12, 15, 18, 21, 16, 19, 22, 25, 28, 31, 34, 37 ]

6.5-2 GAPTableOfMagmaFile

Let file be the name of a file that contains a character table in MAGMA's display format, and identifier be a string. GAPTableOfMagmaFile returns the corresponding GAP character table.

gap> tmpdir:= DirectoryTemporary();;
gap> file:= Filename( tmpdir, "magmatable" );;
gap> str:= "\
> Character Table of Group G\n\
> --------------------------\n\
> \n\
> ---------------------------\n\
> Class |   1  2  3    4    5\n\
> Size  |   1 15 20   12   12\n\
> Order |   1  2  3    5    5\n\
> ---------------------------\n\
> p  =  2   1  1  3    5    4\n\
> p  =  3   1  2  1    5    4\n\
> p  =  5   1  2  3    1    1\n\
> ---------------------------\n\
> X.1   +   1  1  1    1    1\n\
> X.2   +   3 -1  0   Z1 Z1#2\n\
> X.3   +   3 -1  0 Z1#2   Z1\n\
> X.4   +   4  0  1   -1   -1\n\
> X.5   +   5  1 -1    0    0\n\
> \n\
> Explanation of Character Value Symbols\n\
> --------------------------------------\n\
> \n\
> # denotes algebraic conjugation, that is,\n\
> #k indicates replacing the root of unity w by w^k\n\
> \n\
> Z1     = (CyclotomicField(5: Sparse := true)) ! [\n\
> RationalField() | 1, 0, 1, 1 ]\n\
> ";;
gap> FileString( file, str );;
gap> tbl:= GAPTableOfMagmaFile( file, "MagmaA5" );;
gap> Display( tbl );
MagmaA5

     2  2  2  .  .  .
     3  1  .  1  .  .
     5  1  .  .  1  1

       1a 2a 3a 5a 5b
    2P 1a 1a 3a 5b 5a
    3P 1a 2a 1a 5b 5a
    5P 1a 2a 3a 1a 1a

X.1     1  1  1  1  1
X.2     3 -1  .  A *A
X.3     3 -1  . *A  A
X.4     4  .  1 -1 -1
X.5     5  1 -1  .  .

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5
gap> str:= "\
> Character Table of Group G\n\
> --------------------------\n\
> \n\
> ------------------------------\n\
> Class |   1  2   3   4   5   6\n\
> Size  |   1  1   1   1   1   1\n\
> Order |   1  2   3   3   6   6\n\
> ------------------------------\n\
> p  =  2   1  1   4   3   3   4\n\
> p  =  3   1  2   1   1   2   2\n\
> ------------------------------\n\
> X.1   +   1  1   1   1   1   1\n\
> X.2   +   1 -1   1   1  -1  -1\n\
> X.3   0   1  1   J-1-J-1-J   J\n\
> X.4   0   1 -1   J-1-J 1+J  -J\n\
> X.5   0   1  1-1-J   J   J-1-J\n\
> X.6   0   1 -1-1-J   J  -J 1+J\n\
> \n\
> \n\
> Explanation of Character Value Symbols\n\
> --------------------------------------\n\
> \n\
> J = RootOfUnity(3)\n\
> ";;
gap> FileString( file, str );;
gap> tbl:= GAPTableOfMagmaFile( file, "MagmaC6" );;
gap> Display( tbl );
MagmaC6

     2  1  1  1  1   1   1
     3  1  1  1  1   1   1

       1a 2a 3a 3b  6a  6b
    2P 1a 1a 3b 3a  3a  3b
    3P 1a 2a 1a 1a  2a  2a

X.1     1  1  1  1   1   1
X.2     1 -1  1  1  -1  -1
X.3     1  1  A /A  /A   A
X.4     1 -1  A /A -/A  -A
X.5     1  1 /A  A   A  /A
X.6     1 -1 /A  A  -A -/A

A = E(3)
  = (-1+Sqrt(-3))/2 = b3

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