ATLAS: Linear group L_{2}(25)
Order = 7800 = 2^{3}.3.5^{2}.13.
Mult = 2.
Out = 2^{2}.
Standard generators
Standard generators of L_{2}(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.L_{2}(25) = SL_{2}(25)
are preimages A and B where B has order 3 and AB
has order 13.
Standard generators of L_{2}(25):2a = PGL_{2}(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.L_{2}(25).2a =
2.PGL_{2}(25) are preimages C and D where D has
order 3.
Standard generators of L_{2}(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.L_{2}(25):2b are preimages
E and F such that E has order 2 and EF has order 13.
Standard generators of L_{2}(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 L_{2}(25).2^{2} 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 L_{2}(25), L_{2}(25):2a = PGL_{2}(25), L_{2}(25):2b, L_{2}(25).2c and L_{2}(25).2^{2} in terms of their standard generators are given below.
< a, b | a^{2} = b^{3} = (ab)^{13} = [a, b]^{4} = ((ababababab^{-1})^{2}ab^{-1})^{2} = 1 >.
< c, d | c^{2} = d^{3} = (cd)^{8} = [c, dcdcd^{-1}cd(cd^{-1})^{3}cdcd^{-1}cdcd] = 1 >.
< e, f | e^{2} = f^{4} = (ef)^{13} = (ef^{2})^{4} = [e, f]^{5} = [e, fef]^{3} = 1 >.
< g, h | g^{2} = h^{4} = (gh)^{8} = (gh^{2})^{5} = (ghgh^{2})^{3}h^{-3}gh^{2}(ghgh^{-1})^{2}h^{-1}gh^{-1} = 1 >.
< i, j | i^{2} = j^{4} = (ij^{2})^{4} = (ijij^{2})^{3}ij^{-1}ij^{2} = [i, j]^{6} = (ijijij^{2}ijijij^{-1}ij^{2})^{2} = 1 >.
Representations
The representations of L_{2}(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.L_{2}(25) = SL_{2}(25) available are:
- A and
B as
permutations on 208 points.
- A and
B as
2 × 2 matrices over GF(25).
The representations of L_{2}(25):2a = PGL_{2}(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 L_{2}(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 L_{2}(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 L_{2}(25).2^{2} 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 L_{2}(25) are as follows:
The maximal subgroups of L_{2}(25):2a = PGL_{2}(25):2a are as follows:
- L_{2}(25).
- 5^{2}:24.
- D_{52}.
- D_{48}.
The maximal subgroups of L_{2}(25):2b are as follows:
- L_{2}(25).
- 5^{2}:(4 × S_{3}).
- S_{5} × 2.
- S_{5} × 2.
- F_{52} = 13:4.
- D_{8} × S_{3}.
The maximal subgroups of L_{2}(25).2c are as follows:
- L_{2}(25).
- 5^{2}:(3:8).
- F_{52} = 13:4.
- (Q_{8} × 3):2 = 3:SD_{16}.
The maximal subgroups of L_{2}(25).2^{2} are as follows:
- L_{2}(25):2a.
- L_{2}(25):2b.
- L_{2}(25).2c.
- 5^{2}:(24:_{5}2).
- F_{52} × 2 = 26:4.
- (3 × SD_{16}):2 = 3:(8:2^{2}).
Return to main ATLAS page.
Last updated 6th January 1999,
R.A.Wilson, R.A.Parker and J.N.Bray