Stabilizer type Y4

Acting group: 𝔄12
There are exactly 36 groups in this conjugacy class.
The group order is 2.
(T0,S0),
(I,SR).
(T0,S0),
(I,S3R).
(T0,S0),
(I,S5R).
(T0,S0),
(I,S7R).
(T0,S0),
(I,S9R).
(T0,S0),
(I,S11R).
(T0,S0),
(T2I,SR).
(T0,S0),
(T2I,S3R).
(T0,S0),
(T2I,S5R).
(T0,S0),
(T2I,S7R).
(T0,S0),
(T2I,S9R).
(T0,S0),
(T2I,S11R).
(T0,S0),
(T4I,SR).
(T0,S0),
(T4I,S3R).
(T0,S0),
(T4I,S5R).
(T0,S0),
(T4I,S7R).
(T0,S0),
(T4I,S9R).
(T0,S0),
(T4I,S11R).
(T0,S0),
(T6I,SR).
(T0,S0),
(T6I,S3R).
(T0,S0),
(T6I,S5R).
(T0,S0),
(T6I,S7R).
(T0,S0),
(T6I,S9R).
(T0,S0),
(T6I,S11R).
(T0,S0),
(T8I,SR).
(T0,S0),
(T8I,S3R).
(T0,S0),
(T8I,S5R).
(T0,S0),
(T8I,S7R).
(T0,S0),
(T8I,S9R).
(T0,S0),
(T8I,S11R).
(T0,S0),
(T10I,SR).
(T0,S0),
(T10I,S3R).
(T0,S0),
(T10I,S5R).
(T0,S0),
(T10I,S7R).
(T0,S0),
(T10I,S9R).
(T0,S0),
(T10I,S11R).

Where
S=T=(1,2,3,4,5,6,7,8,9,10,11,12) ,
R=(6,7)(5,8)(4,9)(3,10)(2,11)(1,12) ,
I=(7)(6,8)(5,9)(4,10)(3,11)(2,12)(1) ,
F=Q=(10)(8,12)(7)(5,9)(4)(3,11)(2,6)(1) .

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