# ATLAS: Linear group L2(49)

Order = 58800 = 24.3.52.72.
Mult = 2.
Out = 22.

### Standard generators

Standard generators of L2(49) are a and b where a has order 2, b has order 3, ab has order 25 and ababb has order 24.
Standard generators of the double cover 2.L2(49) = SL2(49) are preimages A and B where A has order 3 and AB has order 25.

### Automorphisms

An outer automorphism of type 2a of order 2 may be obtained by mapping (a, b) to (a, bb).

### Presentations

Presentations for L2(49) ... in terms of their standard generators are given below.

< a, b | a2 = b3 = (ababab-1)4 = (ab)6(ab-1)2(ab)3(abab-1)2(ab)4(ab-1)2 = 1 >.

### Representations

The representations of L2(49) available are:
• All primitive permutation representations.
• a and b as permutations on 50 points.
• a and b as permutations on 175 points.
• a and b as permutations on 175 points.
• a and b as permutations on 980 points.
• a and b as permutations on 980 points.
• a and b as permutations on 1176 points.
• a and b as permutations on 1225 points.
• a and b as 3 × 3 matrices over GF(49) - the natural representation as O3(49).
• a and b as 3 × 3 matrices over GF(49).
• a and b as 4 × 4 matrices over GF(7).
The representations of 2.L2(49) = SL2(49) available are:
• A and B as 2 × 2 matrices over GF(49) - the natural representation as SL2(49).
• A and B as 2 × 2 matrices over GF(49).
The representations of L2(49):21 = PGL2(49) available are:
• none.

### Maximal subgroups

The maximal subgroups of L2(49) include the following:
• 72:24.
Order: 1176.
Index: 50.

• PGL2(7).
Order: 336.
Index: 175.

• PGL2(7).
Order: 336.
Index: 175.

• A5.
Order: 60.
Index: 980.

• A5.
Order: 60.
Index: 980.

• D50.
Order: 50.
Index: 1176.

• D48.
Order: 48.
Index: 1225.