The top part of the table lists on the left the primes dividing the group order. Then for each class the
(exponents of ) the prime factorization of the centralizer order are given. (This is redundant information
but useful for some hand calculations.)
M11
The name given to the
table is an just the
stored name - it could
be an arbitrary string
Characters are
arranged according
to degree. The
name X.n simply
stands for χn.
Zeroes in the table
are represented by
dots.
X.1
X.2
X.3
X.4
X.5
X.6
X.7
X.8
X.9
X.10
2
3
5
11
4
2
1
1
4
1
.
.
1
2
.
.
3
.
.
.
.
.
1
.
1
1
.
.
3
.
.
.
3
.
.
.
.
.
.
1
.
.
.
1
2P
3P
5P
11P
1a
1a
1a
1a
1a
2a
1a
2a
2a
2a
3a
3a
1a
3a
3a
4a
2a
4a
4a
4a
5a
5a
5a
1a
5a
6a
3a
2a
6a
6a
8a
4a
8a
8b
8a
8b
4a
8b
8a
8b
11a
11b
11a
11a
1a
11b
11a
11b
11b
1a
1
10
10
10
11
16
16
44
45
55
1 1 1 1 1 1 1
2 1 2 . -1 . .
-2 1 . . 1 A -A
-2 1 . . 1 -A A
3 2 -1 1 . -1 -1
. -2 . 1 . . .
. -2 . 1 . . .
4 -1 . -1 1 . .
-3 . 1 . . -1 -1
-1 1 -1 . -1 1 1
1
-1
-1
-1
.
B
/B
.
1
.
1
-1
-1
-1
.
/B
B
.
1
.
The table header lists
the classes (labelled
by element order
and within each
order by letters of
the alphabet. In
general class
arrangement is
arbitrary though
stored library tables
often arrange
according to
element orders and
class sizes.
The further rows
give for all primes p
dividing the group
order the p-power
maps, i.e. the map
defined by g→gp on
class representatives.
This map indicates
A = E(8)+E(8)^3 = ER(-2) = i2
the cyclic subgroups,
B = E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9 it also can be used to
= (-1+ER(-11))/2 = b11
compute symmetric
and antisymmetric
parts.
Algebraic irrationalities in the table are simply denoted by letters, these letters are explained at the
bottom of the table. Negatives (-), complex conjugates (/) and “unique” Galois conjugates (*) are
denoted by a symbol preceding the letter.
In the explanation E(n) denotes the n-th root of unity e2π/n, i denotes the root of -1. The
abbreviated names (such as ER(-2) or b11) refer to an elaborate naming given in the ATLAS for
certain combinations of roots of unity. The most useful is ER(n), which stands for the square root.