Stabilizer type W'23

Acting group: D12 × Aff1(Z12)
There are exactly 24 groups in this conjugacy class.
The group order is 6.
(T0,S0),
(T4,S4),
(T8,S8),
(I,SR),
(T4I,S5R),
(T8I,S9R).
(T0,S0),
(T4,S4),
(T8,S8),
(I,S3R),
(T4I,S7R),
(T8I,S11R).
(T0,S0),
(T4,S4),
(T8,S8),
(I,S5R),
(T4I,S9R),
(T8I,SR).
(T0,S0),
(T4,S4),
(T8,S8),
(I,S7R),
(T4I,S11R),
(T8I,S3R).
(T0,S0),
(T4,S4),
(T8,S8),
(I,S9R),
(T4I,SR),
(T8I,S5R).
(T0,S0),
(T4,S4),
(T8,S8),
(I,S11R),
(T4I,S3R),
(T8I,S7R).
(T0,S0),
(T4,S4),
(T8,S8),
(T2I,SR),
(T6I,S5R),
(T10I,S9R).
(T0,S0),
(T4,S4),
(T8,S8),
(T2I,S3R),
(T6I,S7R),
(T10I,S11R).
(T0,S0),
(T4,S4),
(T8,S8),
(T2I,S5R),
(T6I,S9R),
(T10I,SR).
(T0,S0),
(T4,S4),
(T8,S8),
(T2I,S7R),
(T6I,S11R),
(T10I,S3R).
(T0,S0),
(T4,S4),
(T8,S8),
(T2I,S9R),
(T6I,SR),
(T10I,S5R).
(T0,S0),
(T4,S4),
(T8,S8),
(T2I,S11R),
(T6I,S3R),
(T10I,S7R).
(T0,S0),
(T4,S8),
(T8,S4),
(I,SR),
(T4I,S9R),
(T8I,S5R).
(T0,S0),
(T4,S8),
(T8,S4),
(I,S3R),
(T4I,S11R),
(T8I,S7R).
(T0,S0),
(T4,S8),
(T8,S4),
(I,S5R),
(T4I,SR),
(T8I,S9R).
(T0,S0),
(T4,S8),
(T8,S4),
(I,S7R),
(T4I,S3R),
(T8I,S11R).
(T0,S0),
(T4,S8),
(T8,S4),
(I,S9R),
(T4I,S5R),
(T8I,SR).
(T0,S0),
(T4,S8),
(T8,S4),
(I,S11R),
(T4I,S7R),
(T8I,S3R).
(T0,S0),
(T4,S8),
(T8,S4),
(T2I,SR),
(T6I,S9R),
(T10I,S5R).
(T0,S0),
(T4,S8),
(T8,S4),
(T2I,S3R),
(T6I,S11R),
(T10I,S7R).
(T0,S0),
(T4,S8),
(T8,S4),
(T2I,S5R),
(T6I,SR),
(T10I,S9R).
(T0,S0),
(T4,S8),
(T8,S4),
(T2I,S7R),
(T6I,S3R),
(T10I,S11R).
(T0,S0),
(T4,S8),
(T8,S4),
(T2I,S9R),
(T6I,S5R),
(T10I,SR).
(T0,S0),
(T4,S8),
(T8,S4),
(T2I,S11R),
(T6I,S7R),
(T10I,S3R).

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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