Stabilizer type W20

Acting group: Aff1(Z12) × D12
There are exactly 6 groups in this conjugacy class.
The group order is 4.
(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).

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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