ATLAS: Symplectic group S6(3)
Order = 4585351680 = 29.39.5.7.13.
Mult = 2.
Out = 2.
Standard generators of S6(3) are a and b where
a is in class 2A, b has order 5, ab has order 13
and abb has order 14.
Standard generators of the double cover 2.S6(3) are preimages A
and B where B has order 5 and AB has order 13.
Standard generators of S6(3):2 are c and d where c is in
class 2A, d is in class 4F, cd has order 20 and
cdd has order 8.
Standard generators of either 2.S6(3).2 are preimages C and D
where CDCDCDCDDCDDD has order 13.
WARNING: The standard generators in v2.0 for S6(3):2 and 2.S6(3).2 were defined on 27.06.00. The representations in v2.0 for S6(3):2 and 2.S6(3).2 are in these standard generators. There are [or were] no standard generators in v1 for S6(3):2 or 2.S6(3).2. The reprsentations of S6(3):2 and 2.S6(3).2 in v1 are NOT in standard generators as we have defined them above.
The representations of S6(3) available are:
-
Permutations on 364 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 1120 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 3640 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 7371 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(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 13 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.S6(3) available are:
-
Dimension 6 over GF(3) - the natural representation:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of S6(3):2 available are:
-
Permutations on 364 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 1120 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 3640 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 7371 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 26 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 13 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 2.S6(3):2 available are:
-
Permutations on 728 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 6 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The maximal subgroups of S6(3) are as follows:
-
31+4:2.U4(2),
with generators here.
-
36:L3(3),
with generators here.
-
31+4:2.(S4 × A4).
-
2.(A4 × U4(2)),
with generators here.
-
22+6:33:S3,
with generators here.
-
L2(27):3,
with generators here.
-
U3(3):2 × 2,
with generators here.
-
L3(3):2,
with generators here.
-
L2(13),
with generators here.
-
L2(13),
with generators here.
-
A5,
with generators here.
The maximal subgroups of S6(3):2 are as follows:
Go to main ATLAS (version 2.0) page.
Go to classical groups page.
Go to old S6(3) page - ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 7th June 2000.
Last updated 02.08.05 by RAW.
Information checked to
Level 0 on 07.06.00 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.