Stabilizer type W'14

Acting group: D12 × Aff1(Z12)
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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