# ATLAS: Linear group L2(25)

Order = 7800 = 23.3.52.13.
Mult = 2.
Out = 22.

### Standard generators

Standard generators of L2(25) are a and b where a has order 2, b has order 3, ab has order 13 and ababb has order 4. These conditions ensure that ab is in class 13A/B.
Standard generators of the double cover 2.L2(25) = SL2(25) are preimages A and B where B has order 3 and AB has order 13.

Standard generators of L2(25):2a = PGL2(25) are c and d where c is in class 2B, d has order 3 and cd has order 8.
Standard generators of either of the double covers 2.L2(25).2a = 2.PGL2(25) are preimages C and D where D has order 3.

Standard generators of L2(25):2b are e and f where e is in class 2C/D, f has order 4 (necessarily class 4B), ef has order 13 and eff has order 4. These conditions imply that ef is in class 13AB.
Standard generators of the double cover 2.L2(25):2b are preimages E and F such that E has order 2 and EF has order 13.

Standard generators of L2(25).2c are g and h where g has order 2, h has order 4 (necessarily class 4C), gh has order 8 and ghh has order 5.

Standard generators of L2(25).22 are i and j where i is in class 2CD, j is in class 4C, ij has order 26 and ijj has order 4.

### Presentations

Presentations for L2(25), L2(25):2a = PGL2(25), L2(25):2b, L2(25).2c and L2(25).22 in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)13 = [a, b]4 = ((ababababab-1)2ab-1)2 = 1 >.

< c, d | c2 = d3 = (cd)8 = [c, dcdcd-1cd(cd-1)3cdcd-1cdcd] = 1 >.

< e, f | e2 = f4 = (ef)13 = (ef2)4 = [e, f]5 = [e, fef]3 = 1 >.

< g, h | g2 = h4 = (gh)8 = (gh2)5 = (ghgh2)3h-3gh2(ghgh-1)2h-1gh-1 = 1 >.

< i, j | i2 = j4 = (ij2)4 = (ijij2)3ij-1ij2 = [i, j]6 = (ijijij2ijijij-1ij2)2 = 1 >.

### Representations

The representations of L2(25) available are:
• All primitive permutation representations.
• a and b as permutations on 26 points.
• a and b as permutations on 65 points.
• a and b as permutations on 65 points - the image of the above under a 2a or 2c outer automorphism.
• a and b as permutations on 300 points.
• a and b as permutations on 325 points.
• a and b as 3 × 3 matrices over GF(25).
• a and b as 13 × 13 matrices over Z (rep 13b).
The representations of 2.L2(25) = SL2(25) available are:
• A and B as permutations on 208 points.
• A and B as 2 × 2 matrices over GF(25).
The representations of L2(25):2a = PGL2(25) available are:
• Permutation representations, including all faithful primitive ones.
• c and d as permutations on 26 points.
• c and d as permutations on 130 points - imprimitive.
• c and d as permutations on 300 points.
• c and d as permutations on 325 points.
The representations of L2(25):2b available are:
• All faithful primitive permutation representations.
• e and f as permutations on 26 points.
• e and f as permutations on 65 points - not yet available.
• e and f as permutations on 65 points - not yet available.
• e and f as permutations on 300 points.
• e and f as permutations on 325 points.
The representations of L2(25).2c available are:
• Permutation representations, including all faithful primitive ones.
• g and h as permutations on 26 points.
• g and h as permutations on 130 points - imprimitive.
• g and h as permutations on 300 points.
• g and h as permutations on 325 points.
The representations of L2(25).22 available are:
• Permutation representations, including all faithful primitive ones.
• i and j as permutations on 26 points.
• i and j as permutations on 130 points - imprimitive.
• i and j as permutations on 300 points.
• i and j as permutations on 325 points.

### Maximal subgroups

The maximal subgroups of L2(25) are as follows:
• 52:12.
• S5.
• S5.
• D26.
• D24.
The maximal subgroups of L2(25):2a = PGL2(25):2a are as follows:
• L2(25).
• 52:24.
• D52.
• D48.
The maximal subgroups of L2(25):2b are as follows:
• L2(25).
• 52:(4 × S3).
• S5 × 2.
• S5 × 2.
• F52 = 13:4.
• D8 × S3.
The maximal subgroups of L2(25).2c are as follows:
• L2(25).
• 52:(3:8).
• F52 = 13:4.
• (Q8 × 3):2 = 3:SD16.
The maximal subgroups of L2(25).22 are as follows:
• L2(25):2a.
• L2(25):2b.
• L2(25).2c.
• 52:(24:52).
• F52 × 2 = 26:4.
• (3 × SD16):2 = 3:(8:22).