Stabilizer type Y41

Acting group: 𝔄12
There are exactly 12 groups in this conjugacy class.
The group order is 4.
(T0,S0),
(T6,S6),
(TI,S4RF),
(T7I,S10RF).
(T0,S0),
(T6,S6),
(TI,S10RF),
(T7I,S4RF).
(T0,S0),
(T6,S6),
(T3I,S4RF),
(T9I,S10RF).
(T0,S0),
(T6,S6),
(T3I,S10RF),
(T9I,S4RF).
(T0,S0),
(T6,S6),
(T3IQ,R),
(T9IQ,S6R).
(T0,S0),
(T6,S6),
(T3IQ,S2R),
(T9IQ,S8R).
(T0,S0),
(T6,S6),
(T3IQ,S4R),
(T9IQ,S10R).
(T0,S0),
(T6,S6),
(T3IQ,S6R),
(T9IQ,R).
(T0,S0),
(T6,S6),
(T3IQ,S8R),
(T9IQ,S2R).
(T0,S0),
(T6,S6),
(T3IQ,S10R),
(T9IQ,S4R).
(T0,S0),
(T6,S6),
(T5I,S4RF),
(T11I,S10RF).
(T0,S0),
(T6,S6),
(T5I,S10RF),
(T11I,S4RF).

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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