# 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.

### Maximal subgroups

The maximal subgroups of Ru are as follows. (Words for generators of maximal subgroups provided by Peter Walsh.)

### Conjugacy class representatives

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.