# ATLAS: Symplectic group S4(9)

Order = 1721606400 = 28.38.52.41.
Mult = 2.
Out = 22.

### Standard generators

Standard generators of S4(9) are a and b where a is in class 2B, b is in class 4B, ab has order 41 and ababbb has order 5.
Standard generators of 2.S4(9) are not yet defined.

Standard generators of the other decorations of S4(9) are not yet defined.

### Black box algorithms

To find standard generators for S4(9):
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 9. If we find such an element, then a is in class 2B. Otherwise, go back to step 1.
3. Find an element of order 8, 12, 20, 24 or 40, and power it up to give an involution t and an element c of order 4.
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 4A, so go back to step 3.
5. Look for a conjugate b of c such that ab has order 41 and abab3 has order 5. If no such conjugate can be found, then c is probably in class 4A, so go back to step 3.
6. The elements a and b are standard generators.

### Representations

The representations of S4(9) available are:
• Permutations on 820[a] points - action on points (Sp4(9)): a and b (GAP).
• Permutations on 820[b] points - action on isotropic lines (Sp4(9)): a and b (GAP).
• Permutations on 1640 points (imprimitive): a and b (GAP).
• Some faithful irreducibles in characteristic 0
• Dimension 41a over Z: a and b (GAP).
• Dimension 41b over Z: a and b (GAP).
The representations of 2.S4(9) = Sp4(9) available are:
• none

Go to main ATLAS (version 2.0) page.
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Version 2.0 created on 21st June 2004.
Last updated 22.06.04 by SJN.