Unitary group U₅(2)

U₅(2): Length 113, 2-generator, 7-relator.

Remark: x and y are R.A.Wilson's standard generators for U₅(2). The presentation is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

• M1 = < x, yxy^-1xy, y²xy^-2xy² > = 2¹⁺⁶:3¹⁺²:2A₄ of order 82 944 and index 165.
• M2 = < x, yxyxy^-2xy > = U₄(2) x 3 of order 77 760 and index 176.
• H2 = < x, yxy^-2xy, y²xy² > = U₄(2) x 3 of order 77 760 and index 176.
• H3 = < x, y²xy^-2, y^-2xy², yxy^-2xy > = U₄(2) x 3 of order 77 760 and index 176.
• H4 = < x, y²xyxy^-2 > = U₄(2) of order 25 920 and index 528.
• M3 = < y, xyxy^-2xyxy^-2xy²xyx > = 2⁴⁺⁴:(3 x A₅) of order 46 080 and index 297. (The coset enumeration over this subgroup proceeds more smoothly if we include the redundant generator xy^-1xy^-1xyxyxyx.)
• M4 = < x, yxyxy^-1 > = 3⁴:S₅ of order 9 720 and index 1 408.

Realisation:
x =
1 0 0 0 0
W 1 0 0 0
1 0 1 0 0
1 0 0 1 0
w 0 0 0 1

y =
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
1 0 0 0 0

Hermitian form: B =
0 w 1 1 W
W 0 w 1 1
1 W 0 w 1
1 1 W 0 w
w 1 1 W 0

These matrices are taken over the field F = GF(4) = { 0, 1, w, W } where W = w² = w+1.

The above matrices and form are available in MAGMA format here. The generators also come in MeatAxe format as x and y .

U₅(2):2: Length ??, ?-generator, ?-relator.

None yet available.