# ATLAS: Linear group L2(17)

Order = 2448 = 24.32.17.
Mult = 2.
Out = 2.

### Standard generators

Standard generators of L2(17) are a and b where a has order 2, b has order 3 and ab has order 17.
Standard generators of the double cover 2.L2(17) = SL2(17) are preimages A and B where B has order 3 and AB has order 17.

Standard generators of L2(17):2 = PGL2(17) are c and d where c is in class 2B, d has order 3, cd has order 16 and cdcdd has order 4. These conditions ensure that cd is in class 16B.
Standard generators of either of the double covers 2.L2(17).2 = 2.PGL2(17) are preimages C and D where D has order 3.

### Presentations

Presentations for L2(17) and L2(17):2 = PGL2(17) in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)17 = ((ab)5(ab-1)3)2 = 1 >.

< c, d | c2 = d3 = (cd)16 = [c, d]4 = [a, (ab)5]2 = 1 >.

### Representations

The representations of L2(17) available are:
• a and b as permutations on 18 points.
• All faithful irreducibles in characteristic 2 and over GF(2).
• a and b as 8 × 8 matrices over GF(2).
• a and b as 8 × 8 matrices over GF(2).
• a and b as 16 × 16 matrices over GF(2).
• a and b as 16 × 16 matrices over GF(8).
• a and b as 16 × 16 matrices over GF(8).
• a and b as 16 × 16 matrices over GF(8).
• a and b as 48 × 48 matrices over GF(2) - reducible over GF(8).
• All faithful irreducibles in characteristic 3.
• a and b as 9 × 9 matrices over GF(9).
• a and b as 9 × 9 matrices over GF(9).
• a and b as 16 × 16 matrices over GF(3).
• a and b as 18 × 18 matrices over GF(3).
• a and b as 18 × 18 matrices over GF(9).
• a and b as 18 × 18 matrices over GF(9).
• All faithful irreducibles in characteristic 17.
• a and b as 3 × 3 matrices over GF(17) - the natural representation as O3(17).
• a and b as 5 × 5 matrices over GF(17).
• a and b as 7 × 7 matrices over GF(17).
• a and b as 9 × 9 matrices over GF(17).
• a and b as 11 × 11 matrices over GF(17).
• a and b as 13 × 13 matrices over GF(17).
• a and b as 15 × 15 matrices over GF(17).
• a and b as 17 × 17 matrices over GF(17).
The representations of 2.L2(17) = SL2(17) available are:
• A and B as 8 × 8 matrices over GF(9).
• A and B as 8 × 8 matrices over GF(9).
• All faithful irreducibles in characteristic 17.
• A and B as 2 × 2 matrices over GF(17) - the natural representation.
• A and B as 4 × 4 matrices over GF(17).
• A and B as 6 × 6 matrices over GF(17).
• A and B as 8 × 8 matrices over GF(17).
• A and B as 10 × 10 matrices over GF(17).
• A and B as 12 × 12 matrices over GF(17).
• A and B as 14 × 14 matrices over GF(17).
• A and B as 16 × 16 matrices over GF(17).
The representations of L2(17):2 = PGL2(17) available are:
• c and d as permutations on 18 points.
• c and d as 3 × 3 matrices over GF(17).

### Maximal subgroups

The maximal subgroups of L2(17) are as follows.
• F136 = 17:8.
• S4.
• S4.
• D18.
• D16.
The maximal subgroups of L2(17):2 = PGL2(17) are as follows.
• L2(17).
• F272 = 17:16.
• D36.
• D32.

### Conjugacy classes

The 11 conjugacy classes of L2(17) are as follows.
• 1A: identity.
• 2A: a.
• 3A: b.
• 4A: [a, bab].
• 8A: (ab)3(ab-1)2.
• 8B: (ab)3ab-1.
• 9A: [a, b].
• 9B: [a, b]2.
• 9C: ababab-1 or [a, b]4.
• 17A: ab.
• 17B: (ab)3.
The 19 conjugacy classes of L2(17):2 = PGL2(17) are as follows.
• 1A: identity.
• 2A: [c, d]2.
• 3A: d.
• 4A: [c, d].
• 8A: (cd)6.
• 8B: (cd)2.
• 9A: (cd)5cd-1.
• 9B: [c, dcdcd].
• 9C: [c, dcd].
• 17AB: (cd)6(cd-1)2.
• 2B: c.
• 6A: (cd)4cd-1.
• 16A: (cd)5.
• 16B: cd.
• 16C: (cd)3.
• 16D: (cd)7.
• 18A: (cd)7(cd-1)2.
• 18B: (cd)3(cd-1)2.
• 18C: cdcdcd-1.