ATLAS: Linear group L_{2}(27)
Order = 9828 = 2^{2}.3^{3}.7.13.
Mult = 2.
Out = 6.
Standard generators
Standard generators of L_{2}(27) are a
and b where
a has order 2, b has order 3 and ab has order 7.
Standard generators of the double cover 2.L_{2}(27) = SL_{2}(27) are preimages A and B where
B has order 3
and AB has order 7.
Standard generators of L_{2}(27).2 are c
and d where
c has order 2 (necessarily 2B),
d has order 4,
and cd has order 7.
Standard generators of the double cover
2.L_{2}(27).2 are preimages C and D where
CD has order 7.
Standard generators of L_{2}(27).3 are e
and f where
e has order 2,
f has order 3,
ef has order 9,
efeff has order 7,
efefefeffefeffeff has order 9
.
Standard generators of the double cover
2.L_{2}(27).3 are preimages E and F where
F has order 3 and
EF has order 9.
Standard generators of L_{2}(27).6 are g
and h where
g has order 2 (necessarily 2B),
h has order 3 (necessarily 3C),
gh has order 24,
and ghghh has order 13.
Standard generators of the double cover
2.L_{2}(27).6 are preimages
G and H where
H has order 3.
Representations
The representations of L_{2}(27) available are

Permutations on 28 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 13 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 13 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 3 over GF(27):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 26 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.L_{2}(27) available are

Dimension 2 over GF(27):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 2 over GF(27):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 2 over GF(27):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
Conjugacy classes
A set of generators for the maximal cyclic subgroups of L2(27) can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
Go to main ATLAS (version 2.0) page.
Go to linear groups page.
Go to old L2(27) page  ATLAS version 1.
Anonymous ftp access is also available on
for.mat.bham.ac.uk.
Version 2.0 file created on 24th January 2002,
from Version 1 file last updated on 06.12.97.
Last updated 27.08.04 by RAW.
Information checked to
Level 0 on 24.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.