# ATLAS: Symplectic group S4(17)

Order = 1004497044480 = 210.34.5.174.29.
Mult = 2.
Out = 2.

### Standard generators

Standard generators of S4(17) are a and b where a is in class 2B, b has order 3, ab has order 145 and ababb has order 5. (This last condition implies that b is in class 3B.)
Standard generators of 2.S4(17) are not yet defined.

Standard generators of S4(17):2 are not yet defined.
Standard generators of 2.S4(17):2 are not yet defined.

### Black box algorithms

#### Finding generators

To find standard generators for S4(17):

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 17. 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 144, and let t=s72, c=s48.
4. Check the order of [t, y] for a few random elements y. If any of these commutators has order greater than 17, 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 145 and ababb has order 5. 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.
This algorithm is available in computer readable format: finder for S4(17).

#### Checking generators

To check that elements x and y of S4(17) are standard generators:

• Check o(x) = 2
• Check o(y) = 3
• Check o(xy) = 145
• Check o(xyxyy) = 5
• Check o([x,yxy]) = 18
This algorithm is available in computer readable format: checker for S4(17).

### Representations

The representations of S4(17) available are:
• Permutations on 5220[a] points - action on points (Sp4(17)): a and b (GAP).
• Permutations on 5220[b] points - action on isotropic lines (Sp4(17)): a and b (GAP).
• Permutations on 10440 points (imprimitive): a and b (GAP).
• Some faithful irreducibles in characteristic 0
• Dimension 290 over Z (reducible over Z(b17)): a and b (GAP).
The representations of 2.S4(17) = Sp4(17) 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 13.1.05 by SJN.