Stabilizer type Z4

Acting group: 𝔇12
There are exactly 36 groups in this conjugacy class.
The group order is 2.
(T0,S0),
(I,SR).
(T0,S0),
(I,S3R).
(T0,S0),
(I,S5R).
(T0,S0),
(I,S7R).
(T0,S0),
(I,S9R).
(T0,S0),
(I,S11R).
(T0,S0),
(T2I,SR).
(T0,S0),
(T2I,S3R).
(T0,S0),
(T2I,S5R).
(T0,S0),
(T2I,S7R).
(T0,S0),
(T2I,S9R).
(T0,S0),
(T2I,S11R).
(T0,S0),
(T4I,SR).
(T0,S0),
(T4I,S3R).
(T0,S0),
(T4I,S5R).
(T0,S0),
(T4I,S7R).
(T0,S0),
(T4I,S9R).
(T0,S0),
(T4I,S11R).
(T0,S0),
(T6I,SR).
(T0,S0),
(T6I,S3R).
(T0,S0),
(T6I,S5R).
(T0,S0),
(T6I,S7R).
(T0,S0),
(T6I,S9R).
(T0,S0),
(T6I,S11R).
(T0,S0),
(T8I,SR).
(T0,S0),
(T8I,S3R).
(T0,S0),
(T8I,S5R).
(T0,S0),
(T8I,S7R).
(T0,S0),
(T8I,S9R).
(T0,S0),
(T8I,S11R).
(T0,S0),
(T10I,SR).
(T0,S0),
(T10I,S3R).
(T0,S0),
(T10I,S5R).
(T0,S0),
(T10I,S7R).
(T0,S0),
(T10I,S9R).
(T0,S0),
(T10I,S11R).

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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