ATLAS: Symplectic group S₄(17)

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.

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

Representations

• 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).

• none

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