ATLAS: Rudvalis group Ru
Order = 145926144000.
Mult = 2.
Out = 1.
Standard generators
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 2Ru are
pre-images A
and B where
A is in class 2B,
B is in class +4A,
and AB has order 13.
Black box algorithms
To find standard generators for Ru:
- Find any element of order 14 or 26. This powers up to a 2B-element, x, say.
- Find any element of order 24. This powers up to a 4A-element, y, say.
- Find a conjugate a of x and a conjugate b of y, whose product has order 13.
Representations
The representations of Ru available are
- a and
b as
28 x 28 matrices over GF(2).
- a and
b as
376 x 376 matrices over GF(2).
- a and
b as
1246 x 1246 matrices over GF(2).
- The other irreducibles in the principal 2-block, 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.
- a and
b as
378 x 378 matrices over GF(9).
- a and
b as
133 x 133 matrices over GF(5).
- a and
b as
273 x 273 matrices over GF(5).
- a and
b as
378 x 378 matrices over GF(5).
- a and
b as
378 x 378 matrices over GF(49).
- a and
b as
378 x 378 matrices over GF(13).
- a and
b as
378 x 378 matrices over GF(29).
- a and
b as
permutations on 4060 points.
The representations of 2Ru available are
- A and
B as
56 x 56 matrices over GF(3).
- A and
B as
28 x 28 matrices over GF(9).
- A and
B as
28 x 28 matrices over GF(5).
- A and
B as
56 x 56 matrices over GF(7).
- A and
B as
28 x 28 matrices over GF(49).
- A and
B as
28 x 28 matrices over GF(13).
- A and
B as
28 x 28 matrices over GF(29).
- A and
B as
permutations on 16240 points.
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).
- 2^6.U3(3).2,
with generators
bb,
(abababb)^-1(abababbababb)^5(abababb).
- (2^2 x Sz(8)):3,
with generators
a,
(ab(ababb)^11)^-1(abababbababb)^5(ab(ababb)^11)
- 2^3+8:L3(2),
with generators
(ababababbababb)^-2bb(ababababbababb)^2,
(abababb)^-1(abababbababb)^5abababb.
- U3(5):2,
with generators
(ababababbababb)^-3bb(ababababbababb)^3,
(abababb)^-1(abababbababb)^5abababb.
- 2^1+4+6S5,
with generators
here.
- L2(25).2.2,
with generators
here.
- A8,
with generators
here.
- L2(29),
with generators
here.
- 5^2:4.S5,
with generators
here.
- 3.A6.2.2,
with generators
here.
- 5^1+2:[2^5],
with generators
here.
- L2(13):2,
with generators
here.
- A6.2.2,
with generators
here.
- 5:4 x A5,
with generators
here (long words, computed by Peter Walsh in about 1994),
or here (shorter words, computed on 9/9/99).
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 29-modular characters, but may not be consistent
with other GAP tables.
- 2A: bb
- 2B: a
- 3A: (babababbababb)^4
- 4A: b
- 4B: b(abababbababb)^3b
- 4C: (abababbababababbababb)^4
- 4D: (bababababbababb)^2
- 5A: (abababb)^2
- 5B: ababbabababbababb
- 6A: (babababbababb)^2
- 7A: (abb)^2
- 8A: (a(bababb)^4a(bababb)^2(bababba)^2(abababb)^2)^3
- 8B: (abababbababababbababb)^2
- 8C: bababababbababb
- 10A: abababb
- 10B: [word too long - try ababababbabbb instead, courtesy of Frank Röhr]
- 12A: (a(bababb)^4a(bababb)^2(bababba)^2(abababb)^2)^2
- 12B: abababbababbb
- 13A: ab
- 14A: abb
- 14B: (abb)^3
- 14C: (abb)^5
- 15A: abababbababb
- 16A: abababbababababbababb
- 16B: (abababbababababbababb)^-1
- 20A: ababababbababb
- 20B: abababbababbababababbababbabb
- 20C: (abababbababbababababbababbabb)^3
- 24A: (a(bababb)^4a(bababb)^2(bababba)^2(abababb)^2)^7
- 24B: a(bababb)^4a(bababb)^2(bababba)^2(abababb)^2
- 26A: ababababbababbabb
- 26B: (ababababbababbabb)^3
- 26C: (ababababbababbabb)^9
- 29A: ababb
- 29B: (ababb)^2
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 are not dealt with in this version.
Return to main ATLAS page.
Last updated 21.12.99
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk