# ATLAS: Symplectic group S4(13)

Order = 68518981440 = 26.32.5.72.134.17.
Mult = 2.
Out = 2.

### Standard generators

Standard generators of S4(13) are a and b where a is in class 2B, b has order 3, ab has order 85 and ababb has order 14. (This last condition implies that b is in class 3B.)
Standard generators of 2.S4(13) are not yet defined.

Standard generators of S4(13):2 are not yet defined.
Standard generators of 2.S4(13):2 are not yet defined.

### Black box algorithms

To find standard generators for S4(13):
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 13. 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 84, and let t=s42, c=s28.
4. Check the order of [t, y] for a few random elements y. If any of these commutators has order greater than 13, 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 85 and ababb has order 14. 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(13) available are:
• Permutations on 2380[a] points - action on points (Sp4(13)): a and b (GAP).
• Permutations on 2380[b] points - action on isotropic lines (Sp4(13)): a and b (GAP).
• Permutations on 4760 points (imprimitive): a and b (GAP).
• Some faithful irreducibles in characteristic 0
• Dimension 170 over Z (reducible over Z(b13)): a and b (GAP).
The representations of 2.S4(13) = Sp4(13) available are:
• none

Go to main ATLAS (version 2.0) page.
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Version 2.0 created on 17th June 2004.
Last updated 17.06.04 by SJN.