# ATLAS: Symplectic group S4(19)

Order = 3057017889600 = 26.34.52.194.181.
Mult = 2.
Out = 2.

### Standard generators

Standard generators of S4(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.S4(19) are not yet defined.

Standard generators of S4(19):2 are not yet defined.
Standard generators of 2.S4(19):2 are not yet defined.

### Black box algorithms

To find standard generators for S4(19):
1. Find an element of even order and power it up to give an involution a.
2. 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.
3. Find an element s of order 180, and let t=s90, c=s60.
4. 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.
5. 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.
6. The elements a and b are standard generators.

### Representations

The representations of S4(19) available are:
• Permutations on 7240[a] points - action on points (Sp4(19)): a and b (GAP).
• Permutations on 7240[b] points - action on isotropic lines (Sp4(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.S4(19) = Sp4(19) available are:
• none Go to main ATLAS (version 2.0) page. Go to classical groups page. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 21st June 2004.
Last updated 21.06.04 by SJN.