Stabilizer type Z12

Acting group: 𝔇12
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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