Stabilizer type X49

Acting group: Aff1(Z12) × Aff1(Z12)
There are exactly 4 groups in this conjugacy class.
The group order is 4.
(T0,S0),
(T6,S4RF),
(T3IQ,S6),
(T9IQ,S10RF).
(T0,S0),
(T6,S4RF),
(T3IQ,S10RF),
(T9IQ,S6).
(T0,S0),
(T6,S10RF),
(T3IQ,S4RF),
(T9IQ,S6).
(T0,S0),
(T6,S10RF),
(T3IQ,S6),
(T9IQ,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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