Acting group: 𝔄12
There are exactly 4 groups in this conjugacy class.
The group order is 2.
(T0,S0),
(IQ,SRF).
(T0,S0),
(IQ,S7RF).
(T0,S0),
(T6IQ,SRF).
(T0,S0),
(T6IQ,S7RF).
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)
.