ATLAS: Symplectic group S_{4}(19)
Order = 3057017889600 = 2^{6}.3^{4}.5^{2}.19^{4}.181.
Mult = 2.
Out = 2.
Standard generators of S_{4}(19) are a and b where
a is in class 2B, b has order 3, ab has order 181 and
ababb has order 6. (This last condition implies that b is
in class 3B.)
Standard generators of 2.S_{4}(19) are not yet defined.
Standard generators of S_{4}(19):2 are not yet defined.
Standard generators of 2.S_{4}(19):2 are not yet defined.
To find standard generators for S_{4}(19):
 Find an element of even order and power it up to give an involution
a.
 Look for an element z such that [a, z] has
order greater than 19. If we find such an element,
then a is in class 2B. Otherwise, go back to step 1.
 Find an element s of order 180, and
let t=s^{90}, c=s^{60}.
 Check the order of [t, y] for a few random
elements y.
If any of these commutators has order greater than 19, then
c is in class 3A, so go back to step 3.
 Look for a conjugate b of c such that ab has
order 181 and ababb has order 6. If no such conjugate can be
found, then c is probably in class 3A, so go back to step 3.
 The elements a and b are standard generators.
The representations of S_{4}(19) available are:

Permutations on 7240[a] points  action on points (Sp_{4}(19)):
a and
b (GAP).

Permutations on 7240[b] points  action on isotropic lines (Sp_{4}(19)):
a and
b (GAP).

Permutations on 14480 points (imprimitive):
a and
b (GAP).
 Some faithful irreducibles in characteristic 0
 Dimension 362 over Z (reducible over Z(b19)):
a and b (GAP).
The representations of 2.S_{4}(19) = Sp_{4}(19) available are:
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See here for details.
Version 2.0 created on 21st June 2004.
Last updated 21.06.04 by SJN.