COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

114 J. S. Frame
Ho of order 29345*7 whose 30 classes and characters were described by
Frame [4]. The 60 classes of H are labeled 1, l’, 2, 2’, . . . , 30, 30’, so that
class k’ contains the product by -I of an element of class k. There are
46 classes of A containing elements of H. For the 12 classes of A that split
in H the corresponding classes of Hand the values of the induced permutation
character 120, are:
(2.1)
A
H 120, A H
i I
1422 2, 16 30+2 162
122 4 5,18,19 6+1+1 1223
1 223 8,21 6+2 122 2-2
1223 24,27 1+1 132 3
l’, 16’
2’, 3’,
17’
4’, 28’
6’, 21’
120, A I H 1 120,
l-l-63 12123 7’,8’ l-i-3
3+1+12 232 9’, 22’ 1+3
lf3 126 lo’, 23’ 1+3
1+15 125 15’, 25’ 1 + 3
(2) Twelve additional classes of A that are not represented in H but are
represented in the monomial subgroup M’ = M/C of A = F/C are denoted
by the symbols
--
I 22 3, 4zu, 24, 42, i i 3 3~, 2 Bv, 1 3 4, I 2 5, 2 3 3v, 224, 1 i 6, 8#.
(2.2)
(3) Five of the remaining nine classes of A contain 8 X 8 real orthogonal
symplectic matrices that commute with a skew matrix of type Z4 and order
4 that we call “2’. By appropriate choice of “i” the 8 X 8 orthogonal matrices
are equivalent to 4X4 unitary matrices A+ iB under the correspondence
0 z A B
i - AfiB- -B A
[ -z 0 1 ’ [ 1 (2.3)
The 210325 matrices of this type form the normalizer Nj of i, which has a
monomial subgroup Mi of order 2’4! and index 15. In six cosets of Mi
the matrices have exactly 2 zero entries per row or column. In the other
eight cosets there are no zero entries.
The classes of type 1214, its square s2/j2, and fourth power 34/r4 are
represented by the following unitary matrices of orders 24, 12, 6 in F, or
12, 6 and 3 in A. Roman numerals indicate Hamill’s class symbols [7]
.-~~:l-~-~~~:,I),tl~~I’jIII; -; 1; I//
0 l-i i i i -i 1 -1 1 11
Type 1214
(LXVII)
Type g2/Z2
(LXW
Type g4/14
(LW
(2.4)
The characters of the Weyl group Es 115
- -
Type 1012 and its square j2/i2 belong to classes of elements of orders
20 and 10 in F, or 10 and 5 in A = F/C. Representative 4X4 unitary
matrices A+ iB for these classes are
-I
0 Ifi 0 l-i
1 1 i il
1 l- i 0 l+i 0 I 1 -1 1 i-i
. (2.5)
2 -1+ i 0 lfi 0 ’ z -1 -1 i i I
1 O
lfi 0 -1fiJ l - l i -ij
- -
Type 1012
Type S2/T2
(LXIV)
(LX)
(4) The four remaining classes are Miss Hamill’s classes LVII, LVIII,
LIX, and LXV [7], here denoted 62/22v, 9/l, g2 6/12 2, and 1 3-l 5-l 15
respectively, and represented by the orthogonal matrices
r l-l 3 l-l-l-l-l r-1-3 1 1 1 1 l-l
l-l-l 1 3-l-l-l -1 l-3 I 1 1 l-l
-1 1 I 3 1 1 1 1 -1 1 l-3 1 1 l-l
1 -3-l-l l-l-l-l-l _1_ -1 1 1 l-3 1 l-l
-z -1-3 l-l 1 1 1 1 ’ 4 -1 1 1 1 l-3 l-l
l-l-l l-l-l-l 3 -1 1 1 1 1 -3-l
l-l-l l-l 3-l-l -1 1 1 1 1 1 1 3
l-l-l l-l-l 3-l L 3 1 1 1 1 1 l-1
_- _
Type 62/22v
(LVII)
l-3 1 1 1 1 1 1
1 l-3 1 1 1 1 1
1 1 1 l-3 1 1 1
-1-l -1 3 -1-l -1-l
3 -1-l -1-I -1-l -1
11 11 1 l-3 1
11 1 1 1 1 l-2
1 1 1 1 l-3 1 1
Type 5” s/i2 2
(LW
Type 9/l
(LVIII)
(2.6)
-1-1-1 0 0 0 1 0
1 1-l 0 0 0 1 0
0 0 o-1-1-1 0 1
l-l-l 0 0 o-1 0
0 0 O-l-l 1 o-1
0 0 0 l-l-l o-1
0 0 o-1 l-l o-1
1 l-l 10 0 0 1 0
Type 1 3-l 5-l 15
G-XV)
--.
Type 9/l IS the negative of a 9-cycle of SQ, represented in F by the negative
of the product of R in (1 .l) by an 8-cycle permutation matrix of M
in which the signs are changed to negative in the last row and column.
The negative of a matrix of type 13-r 5-r 15 represents an element of order
30 in F whose 5th power is of type 34/i4.
,