ATLAS: Rudvalis group Ru
Order = 145926144000 = 214.33.53.7.13.29.
Mult = 2.
Out = 1.
The following information is available for Ru:
Standard generators of the Rudvalis group Ru are a
and b where a is in class 2B, b is in class 4A
and ab has order 13.
Standard generators of the double cover 2.Ru are preimages A
and B where B is in class +4A and AB has order 13. An
equivalent condition to B being in class +4A is that ABABB has
order 29.
Finding generators
To find standard generators for Ru:
- Find any element of order 14 or 26. This powers up to x in class 2B.
[The probability of success at each attempt is 15 in 91 (about 1 in 6).]
-
Find any element of order 24. This powers up to y in class 4A.
[The probability of success at each attempt is 1 in 12.]
-
Find a conjugate a of x and a conjugate b of y, whose product has order 13.
[The probability of success at each attempt is 8 in 1305 (about 1 in 163).]
-
Now a and b are standard generators of Ru.
This algorithm is available in computer readable format:
finder for Ru.
Checking generators
To check that elements x and y of Ru
are standard generators:
- Check o(x) = 2
- Check o(y) = 4
- Check o(xy) = 13
- Check o(xyy) = 14
- Check o(xyxyy) = 29
This algorithm is available in computer readable format:
checker for Ru.
The representations of Ru available are:
-
Permutations on 4060 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Faithful irreducibles in characteristic 2.
-
Dimension 28 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 376 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 1246 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary).
-
The other irreducibles in the principal 2block, of degrees
7280 and 16036, have been constructed. As they are rather large,
they are not in the main ATLAS. Please send an
email if you want them.
- Faithful irreducibles in characteristic 3.
-
Dimension 378 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 406 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 783 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Faithful irreducibles in characteristic 5.
-
Dimension 133 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 273 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 378 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 783 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 378 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 406 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 782 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 378 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 406 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 783 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 378 over GF(29):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 406 over GF(29):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 783 over GF(29):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.Ru available are:
-
Permutations on 16240 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 56 over GF(3) - reducible over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 28 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 28 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 912 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 56 over GF(7) - reducible over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 28 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 28 over GF(13):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 28 over GF(29):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The maximal subgroups of Ru are as follows.
Words for generators of maximal subgroups provided by Peter Walsh.
-
2F4(2) = 2F4(2)'.2, with generators
bb, (abababb(abababbababb)3)-1b(abababb(abababbababb)3).
-
26.U3(3).2,
with generators
bb,
(abababb)-1(abababbababb)5(abababb).
-
(22 × Sz(8)):3,
with generators
a,
(ab(ababb)11)-1(abababbababb)5(ab(ababb)11).
-
23+8:L3(2),
with generators
(ababababbababb)-2bb(ababababbababb)2,
(abababb)-1(abababbababb)5(abababb).
-
U3(5):2,
with generators
(ababababbababb)-3bb(ababababbababb)3,
(abababb)-1(abababbababb)5(abababb).
-
21+4+6.S5,
with generators
ababababbababb,
((ababababbababb)10aababb)6((ababababbababb)10aababb(abababbab)^2)4.
-
L2(25).22,
with generators
here.
-
A8,
with generators
here.
-
L2(29),
with generators
(ababb)-16a(ababb)16,
(ab)-3(abababbababb)5(ab)3.
-
52:4.S5,
with generators
here.
-
3.A6.22,
with generators
here.
-
51+2:[25],
with generators
here.
-
L2(13):2,
with generators
(ababb)-2a(ababb)2,
(ab)-4(abababbababb)5(ab)4.
-
A6.22,
with generators
here.
-
5:4 × A5,
with generators
here.
Some [shorter] words that have subsequently been computed.
-
2F4(2) = 2F4(2)'.2, with generators
b2, abababab2aba.
-
26.U3(3).2,
with generators
b2,
abab3abab2ab.
-
(22 × Sz(8)):3,
with generators
a,
ab2abab2abab2ab3.
-
23+8:L3(2),
with generators
b2,
abab2abab3ab2abab3.
-
U3(5):2,
with generators
b2,
abab3ab3(ab2)5.
-
21+4+6.S5,
with generators
b2,
abab2ab3ab(ab2)4.
-
L2(25).22,
with generators
b2,
abababab2ab3abab.
-
A8,
with generators
b2,
(ab)5ab3abab2ab3ab.
-
L2(29), with generators
a, ababab3ab3ab2ab3ab3abab,
and standard generators
aabbabbabab,
(ab)4ab2(ab3)3.
We obtain standard generators of L2(29) in terms of the
generators (x, y) = (a,
ababab3ab3ab2ab3ab3abab) as x, xy12
-
52:4.S5,
with generators
b2,
abab2ab3(ab)4ab2ab2aba.
-
3.A6.2.2,
with generators
a,
(ab2)3ababab2ab3ab(ab2)3.
-
51+2:[25],
with generators
a,
(abab3ab)2abab2ab3ab2.
-
L2(13):2,
with generators
a,
ababab3ab3(ab2abab3)2.
-
A6.22,
with generators
a,
(ab2)3ab(ab2)2(ab2ab3)2ab.
-
5:4 × A5,
with generators
here, and with generators
a,
(abab2)2ab3ab2(ab3)2(ab2)2abab2.
The following conjugacy class representatives have been computed by
Peter Walsh. Click here for program to compute them.
The choice of classes among algebraic conjugates is arbitrary, consistent
with the ordinary ATLAS. The given choice has been used by
Frank Röhr in calculating the 13 and 29modular characters, but
may not be consistent with other GAP tables. (Actually, we have replaced
some of Peter's class representatives by shorter words while retaining his
definitions of the classes. Version 1 contains Peter's class list.)
- 1A: identity [or a2].
- 2A: b2.
- 2B: a.
- 3A: (ababab2abab3)4 or (ababab2abab2)5 or (ab)4ab2(ab3)3.
- 4A: b.
- 4B: (ababab2abab3)3.
- 4C: (abab2ab3ab)2b.
- 4D: [b2, aba].
- 5A: (ababab2)2.
- 5B: ab(ab2)4.
- 6A: (ababab2abab3)2 or (abab2)2(bab)2.
- 7A: (ab2)2 or [a, b].
- 8A: (abab3)2(ab2)3 or abab(ab2)4ab3.
- 8B: b2abababa.
- 8C: (abab3)2ab2.
- 10A: ababab2.
- 10B: [word too long - try abababab2ab3 instead, courtesy of Frank Röhr.]
- 12A: ab(ab2)3ab3ab2.
- 12B: ababab2abab3.
- 13A: ab.
- 14A: ab2.
- 14B: (ab2)3.
- 14C: (ab2)5 or ab(ab2)3.
- 15A: ababab2abab2.
- 16A: ababab2abababab2abab2 or abababab2ab2ab3.
- 16B: (ababab2abababab2abab2)-1 or abababab3ab2ab2.
- 20A: abababab3 or ababab2ab2 or abababab2abab2.
- 20B: ababab2abab2abababab2abab2ab2 or (ababab3)2ab2.
- 20C: (ababab2abab2abababab2abab2ab2)3 or ab(ab2)4ab3ab2.
- 24A: (a(babab2)4a(babab2)2(babab2a)2(ababab2)2)7 or (ab)4ab2ab2ab3.
- 24B: a(babab2)4a(babab2)2(babab2a)2(ababab2)2 or abab(abab3)2ab3.
- 26A: abababab2abab2ab2 or [a, bab2].
- 26B: (abababab2abab2ab2)3 or abab2ab3.
- 26C: (abababab2abab2ab2)9 or ababababab3.
- 29A: abab2.
- 29B: (abab2)2 or ababab3.
A different set of generators for the maximal cyclic subgroups
(up to conjugacy) can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
Problems of algebraic conjugacy: all classes except 14ABC and 20BC are
compatible with the other class list; 14ABC and 20BC are not dealt with in this version.
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old Ru page - ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 21st May 1999.
Last updated 10.1.05 by SJN.
Information checked to
Level 0 on 21.05.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.