ATLAS: Linear group L_{2}(16)
Order = 4080 = 2^{4}.3.5.17.
Mult = 1.
Out = 4.
Standard generators
Standard generators of L_{2}(16) are a and b where
a has order 2, b has order 3 and ab has order 15.
Standard generators of L_{2}(16):2 are c and d where c is in class 2B, d has order 4 and cd has order 15.
Standard generators of L_{2}(16):4 are e and f where e is in class 2A, f is in class 4B/B', ef has order 8, eff has order 6 and efeffefff has order 4.
.
Presentations
Presentations for L_{2}(16), L_{2}(16):2 and L_{2}(16):4 in terms of their standard generators are given below.
< a, b | a^{2} = b^{3} = (ab)^{15} = ((ab)^{5}(ab^{-1})^{3})^{2} = 1 >.
< c, d | c^{2} = d^{4} = [c, dcd]^{2} = cd(cdcd^{2})^{2}(cd^{2}cd)^{2}cdcd^{-1} = 1 >.
< e, f | e^{2} = f^{4} = (ef)^{8} = (ef^{2})^{6} = (ef)^{3}(ef^{-1})^{3}(ef)^{2}(ef^{-1})^{2}(ef)^{2}ef^{2} = 1 >.
Representations
The representations of L_{2}(16) available are
- a and
b as
permutations on 17 points.
- a and
b as
2 × 2 matrices over GF(16) - the natural representation.
Maximal subgroups
The maximal subgroups of L_{2}(16) are as follows.
- 2^{4}:15._{ }
- A_{5}.
- D_{34}.
- D_{30}.
The maximal subgroups of L_{2}(16):2 are as follows.
- L_{2}(16).
- 2^{4}:(3 × D_{10}).
- A_{5} × 2.
- F_{68} = 17:4.
- D_{10} × S_{3}.
The maximal subgroups of L_{2}(16):4 are as follows.
- L_{2}(16):2.
- 2^{4}:15:4._{ }
- (A_{5} × 2).2.
- F_{136} = 17:8.
- 5:4 × S_{3}.
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Last updated 8th October 1998,
R.A.Wilson, R.A.Parker and J.N.Bray