Stabilizer type X89

Acting group: Aff1(Z12) × Aff1(Z12)
There are exactly 24 groups in this conjugacy class.
The group order is 24.
(T0,S0),
(T2,RF),
(T4,S4),
(T6,S4RF),
(T8,S8),
(T10,S8RF),
(TI,S2F),
(T3I,S2R),
(T5I,S6F),
(T7I,S6R),
(T9I,S10F),
(T11I,S10R),
(Q,S4F),
(T2Q,S4R),
(T4Q,S8F),
(T6Q,S8R),
(T8Q,F),
(T10Q,R),
(TIQ,S10),
(T3IQ,S10RF),
(T5IQ,S2),
(T7IQ,S2RF),
(T9IQ,S6),
(T11IQ,S6RF).
(T0,S0),
(T2,RF),
(T4,S4),
(T6,S4RF),
(T8,S8),
(T10,S8RF),
(TI,S2R),
(T3I,S6F),
(T5I,S6R),
(T7I,S10F),
(T9I,S10R),
(T11I,S2F),
(Q,F),
(T2Q,R),
(T4Q,S4F),
(T6Q,S4R),
(T8Q,S8F),
(T10Q,S8R),
(TIQ,S2RF),
(T3IQ,S6),
(T5IQ,S6RF),
(T7IQ,S10),
(T9IQ,S10RF),
(T11IQ,S2).
(T0,S0),
(T2,RF),
(T4,S4),
(T6,S4RF),
(T8,S8),
(T10,S8RF),
(TI,S6F),
(T3I,S6R),
(T5I,S10F),
(T7I,S10R),
(T9I,S2F),
(T11I,S2R),
(Q,S8F),
(T2Q,S8R),
(T4Q,F),
(T6Q,R),
(T8Q,S4F),
(T10Q,S4R),
(TIQ,S10),
(T3IQ,S10RF),
(T5IQ,S2),
(T7IQ,S2RF),
(T9IQ,S6),
(T11IQ,S6RF).
(T0,S0),
(T2,RF),
(T4,S4),
(T6,S4RF),
(T8,S8),
(T10,S8RF),
(TI,S6R),
(T3I,S10F),
(T5I,S10R),
(T7I,S2F),
(T9I,S2R),
(T11I,S6F),
(Q,S4F),
(T2Q,S4R),
(T4Q,S8F),
(T6Q,S8R),
(T8Q,F),
(T10Q,R),
(TIQ,S2RF),
(T3IQ,S6),
(T5IQ,S6RF),
(T7IQ,S10),
(T9IQ,S10RF),
(T11IQ,S2).
(T0,S0),
(T2,RF),
(T4,S4),
(T6,S4RF),
(T8,S8),
(T10,S8RF),
(TI,S10F),
(T3I,S10R),
(T5I,S2F),
(T7I,S2R),
(T9I,S6F),
(T11I,S6R),
(Q,F),
(T2Q,R),
(T4Q,S4F),
(T6Q,S4R),
(T8Q,S8F),
(T10Q,S8R),
(TIQ,S10),
(T3IQ,S10RF),
(T5IQ,S2),
(T7IQ,S2RF),
(T9IQ,S6),
(T11IQ,S6RF).
(T0,S0),
(T2,RF),
(T4,S4),
(T6,S4RF),
(T8,S8),
(T10,S8RF),
(TI,S10R),
(T3I,S2F),
(T5I,S2R),
(T7I,S6F),
(T9I,S6R),
(T11I,S10F),
(Q,S8F),
(T2Q,S8R),
(T4Q,F),
(T6Q,R),
(T8Q,S4F),
(T10Q,S4R),
(TIQ,S2RF),
(T3IQ,S6),
(T5IQ,S6RF),
(T7IQ,S10),
(T9IQ,S10RF),
(T11IQ,S2).
(T0,S0),
(T2,S2RF),
(T4,S8),
(T6,S10RF),
(T8,S4),
(T10,S6RF),
(TI,R),
(T3I,S6F),
(T5I,S8R),
(T7I,S2F),
(T9I,S4R),
(T11I,S10F),
(Q,F),
(T2Q,S2R),
(T4Q,S8F),
(T6Q,S10R),
(T8Q,S4F),
(T10Q,S6R),
(TIQ,RF),
(T3IQ,S6),
(T5IQ,S8RF),
(T7IQ,S2),
(T9IQ,S4RF),
(T11IQ,S10).
(T0,S0),
(T2,S2RF),
(T4,S8),
(T6,S10RF),
(T8,S4),
(T10,S6RF),
(TI,S2F),
(T3I,S4R),
(T5I,S10F),
(T7I,R),
(T9I,S6F),
(T11I,S8R),
(Q,F),
(T2Q,S2R),
(T4Q,S8F),
(T6Q,S10R),
(T8Q,S4F),
(T10Q,S6R),
(TIQ,S2),
(T3IQ,S4RF),
(T5IQ,S10),
(T7IQ,RF),
(T9IQ,S6),
(T11IQ,S8RF).
(T0,S0),
(T2,S2RF),
(T4,S8),
(T6,S10RF),
(T8,S4),
(T10,S6RF),
(TI,S4R),
(T3I,S10F),
(T5I,R),
(T7I,S6F),
(T9I,S8R),
(T11I,S2F),
(Q,S4F),
(T2Q,S6R),
(T4Q,F),
(T6Q,S2R),
(T8Q,S8F),
(T10Q,S10R),
(TIQ,RF),
(T3IQ,S6),
(T5IQ,S8RF),
(T7IQ,S2),
(T9IQ,S4RF),
(T11IQ,S10).
(T0,S0),
(T2,S2RF),
(T4,S8),
(T6,S10RF),
(T8,S4),
(T10,S6RF),
(TI,S6F),
(T3I,S8R),
(T5I,S2F),
(T7I,S4R),
(T9I,S10F),
(T11I,R),
(Q,S4F),
(T2Q,S6R),
(T4Q,F),
(T6Q,S2R),
(T8Q,S8F),
(T10Q,S10R),
(TIQ,S2),
(T3IQ,S4RF),
(T5IQ,S10),
(T7IQ,RF),
(T9IQ,S6),
(T11IQ,S8RF).
(T0,S0),
(T2,S2RF),
(T4,S8),
(T6,S10RF),
(T8,S4),
(T10,S6RF),
(TI,S8R),
(T3I,S2F),
(T5I,S4R),
(T7I,S10F),
(T9I,R),
(T11I,S6F),
(Q,S8F),
(T2Q,S10R),
(T4Q,S4F),
(T6Q,S6R),
(T8Q,F),
(T10Q,S2R),
(TIQ,RF),
(T3IQ,S6),
(T5IQ,S8RF),
(T7IQ,S2),
(T9IQ,S4RF),
(T11IQ,S10).
(T0,S0),
(T2,S2RF),
(T4,S8),
(T6,S10RF),
(T8,S4),
(T10,S6RF),
(TI,S10F),
(T3I,R),
(T5I,S6F),
(T7I,S8R),
(T9I,S2F),
(T11I,S4R),
(Q,S8F),
(T2Q,S10R),
(T4Q,S4F),
(T6Q,S6R),
(T8Q,F),
(T10Q,S2R),
(TIQ,S2),
(T3IQ,S4RF),
(T5IQ,S10),
(T7IQ,RF),
(T9IQ,S6),
(T11IQ,S8RF).
(T0,S0),
(T2,S6RF),
(T4,S4),
(T6,S10RF),
(T8,S8),
(T10,S2RF),
(TI,R),
(T3I,S10F),
(T5I,S4R),
(T7I,S2F),
(T9I,S8R),
(T11I,S6F),
(Q,S4F),
(T2Q,S10R),
(T4Q,S8F),
(T6Q,S2R),
(T8Q,F),
(T10Q,S6R),
(TIQ,S8RF),
(T3IQ,S6),
(T5IQ,RF),
(T7IQ,S10),
(T9IQ,S4RF),
(T11IQ,S2).
(T0,S0),
(T2,S6RF),
(T4,S4),
(T6,S10RF),
(T8,S8),
(T10,S2RF),
(TI,S2F),
(T3I,S8R),
(T5I,S6F),
(T7I,R),
(T9I,S10F),
(T11I,S4R),
(Q,S4F),
(T2Q,S10R),
(T4Q,S8F),
(T6Q,S2R),
(T8Q,F),
(T10Q,S6R),
(TIQ,S10),
(T3IQ,S4RF),
(T5IQ,S2),
(T7IQ,S8RF),
(T9IQ,S6),
(T11IQ,RF).
(T0,S0),
(T2,S6RF),
(T4,S4),
(T6,S10RF),
(T8,S8),
(T10,S2RF),
(TI,S4R),
(T3I,S2F),
(T5I,S8R),
(T7I,S6F),
(T9I,R),
(T11I,S10F),
(Q,S8F),
(T2Q,S2R),
(T4Q,F),
(T6Q,S6R),
(T8Q,S4F),
(T10Q,S10R),
(TIQ,S8RF),
(T3IQ,S6),
(T5IQ,RF),
(T7IQ,S10),
(T9IQ,S4RF),
(T11IQ,S2).
(T0,S0),
(T2,S6RF),
(T4,S4),
(T6,S10RF),
(T8,S8),
(T10,S2RF),
(TI,S6F),
(T3I,R),
(T5I,S10F),
(T7I,S4R),
(T9I,S2F),
(T11I,S8R),
(Q,S8F),
(T2Q,S2R),
(T4Q,F),
(T6Q,S6R),
(T8Q,S4F),
(T10Q,S10R),
(TIQ,S10),
(T3IQ,S4RF),
(T5IQ,S2),
(T7IQ,S8RF),
(T9IQ,S6),
(T11IQ,RF).
(T0,S0),
(T2,S6RF),
(T4,S4),
(T6,S10RF),
(T8,S8),
(T10,S2RF),
(TI,S8R),
(T3I,S6F),
(T5I,R),
(T7I,S10F),
(T9I,S4R),
(T11I,S2F),
(Q,F),
(T2Q,S6R),
(T4Q,S4F),
(T6Q,S10R),
(T8Q,S8F),
(T10Q,S2R),
(TIQ,S8RF),
(T3IQ,S6),
(T5IQ,RF),
(T7IQ,S10),
(T9IQ,S4RF),
(T11IQ,S2).
(T0,S0),
(T2,S6RF),
(T4,S4),
(T6,S10RF),
(T8,S8),
(T10,S2RF),
(TI,S10F),
(T3I,S4R),
(T5I,S2F),
(T7I,S8R),
(T9I,S6F),
(T11I,R),
(Q,F),
(T2Q,S6R),
(T4Q,S4F),
(T6Q,S10R),
(T8Q,S8F),
(T10Q,S2R),
(TIQ,S10),
(T3IQ,S4RF),
(T5IQ,S2),
(T7IQ,S8RF),
(T9IQ,S6),
(T11IQ,RF).
(T0,S0),
(T2,S8RF),
(T4,S8),
(T6,S4RF),
(T8,S4),
(T10,RF),
(TI,S2F),
(T3I,S10R),
(T5I,S10F),
(T7I,S6R),
(T9I,S6F),
(T11I,S2R),
(Q,F),
(T2Q,S8R),
(T4Q,S8F),
(T6Q,S4R),
(T8Q,S4F),
(T10Q,R),
(TIQ,S2),
(T3IQ,S10RF),
(T5IQ,S10),
(T7IQ,S6RF),
(T9IQ,S6),
(T11IQ,S2RF).
(T0,S0),
(T2,S8RF),
(T4,S8),
(T6,S4RF),
(T8,S4),
(T10,RF),
(TI,S2R),
(T3I,S2F),
(T5I,S10R),
(T7I,S10F),
(T9I,S6R),
(T11I,S6F),
(Q,S8F),
(T2Q,S4R),
(T4Q,S4F),
(T6Q,R),
(T8Q,F),
(T10Q,S8R),
(TIQ,S6RF),
(T3IQ,S6),
(T5IQ,S2RF),
(T7IQ,S2),
(T9IQ,S10RF),
(T11IQ,S10).
(T0,S0),
(T2,S8RF),
(T4,S8),
(T6,S4RF),
(T8,S4),
(T10,RF),
(TI,S6F),
(T3I,S2R),
(T5I,S2F),
(T7I,S10R),
(T9I,S10F),
(T11I,S6R),
(Q,S4F),
(T2Q,R),
(T4Q,F),
(T6Q,S8R),
(T8Q,S8F),
(T10Q,S4R),
(TIQ,S2),
(T3IQ,S10RF),
(T5IQ,S10),
(T7IQ,S6RF),
(T9IQ,S6),
(T11IQ,S2RF).
(T0,S0),
(T2,S8RF),
(T4,S8),
(T6,S4RF),
(T8,S4),
(T10,RF),
(TI,S6R),
(T3I,S6F),
(T5I,S2R),
(T7I,S2F),
(T9I,S10R),
(T11I,S10F),
(Q,F),
(T2Q,S8R),
(T4Q,S8F),
(T6Q,S4R),
(T8Q,S4F),
(T10Q,R),
(TIQ,S6RF),
(T3IQ,S6),
(T5IQ,S2RF),
(T7IQ,S2),
(T9IQ,S10RF),
(T11IQ,S10).
(T0,S0),
(T2,S8RF),
(T4,S8),
(T6,S4RF),
(T8,S4),
(T10,RF),
(TI,S10F),
(T3I,S6R),
(T5I,S6F),
(T7I,S2R),
(T9I,S2F),
(T11I,S10R),
(Q,S8F),
(T2Q,S4R),
(T4Q,S4F),
(T6Q,R),
(T8Q,F),
(T10Q,S8R),
(TIQ,S2),
(T3IQ,S10RF),
(T5IQ,S10),
(T7IQ,S6RF),
(T9IQ,S6),
(T11IQ,S2RF).
(T0,S0),
(T2,S8RF),
(T4,S8),
(T6,S4RF),
(T8,S4),
(T10,RF),
(TI,S10R),
(T3I,S10F),
(T5I,S6R),
(T7I,S6F),
(T9I,S2R),
(T11I,S2F),
(Q,S4F),
(T2Q,R),
(T4Q,F),
(T6Q,S8R),
(T8Q,S8F),
(T10Q,S4R),
(TIQ,S6RF),
(T3IQ,S6),
(T5IQ,S2RF),
(T7IQ,S2),
(T9IQ,S10RF),
(T11IQ,S10).

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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