Stabilizer type Y37

Acting group: 𝔄12
There are exactly 18 groups in this conjugacy class.
The group order is 4.
(T0,S0),
(T6,S6),
(TI,R),
(T7I,S6R).
(T0,S0),
(T6,S6),
(TI,S2R),
(T7I,S8R).
(T0,S0),
(T6,S6),
(TI,S4R),
(T7I,S10R).
(T0,S0),
(T6,S6),
(TI,S6R),
(T7I,R).
(T0,S0),
(T6,S6),
(TI,S8R),
(T7I,S2R).
(T0,S0),
(T6,S6),
(TI,S10R),
(T7I,S4R).
(T0,S0),
(T6,S6),
(T3I,R),
(T9I,S6R).
(T0,S0),
(T6,S6),
(T3I,S2R),
(T9I,S8R).
(T0,S0),
(T6,S6),
(T3I,S4R),
(T9I,S10R).
(T0,S0),
(T6,S6),
(T3I,S6R),
(T9I,R).
(T0,S0),
(T6,S6),
(T3I,S8R),
(T9I,S2R).
(T0,S0),
(T6,S6),
(T3I,S10R),
(T9I,S4R).
(T0,S0),
(T6,S6),
(T5I,R),
(T11I,S6R).
(T0,S0),
(T6,S6),
(T5I,S2R),
(T11I,S8R).
(T0,S0),
(T6,S6),
(T5I,S4R),
(T11I,S10R).
(T0,S0),
(T6,S6),
(T5I,S6R),
(T11I,R).
(T0,S0),
(T6,S6),
(T5I,S8R),
(T11I,S2R).
(T0,S0),
(T6,S6),
(T5I,S10R),
(T11I,S4R).

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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