# Properties

 Label 12T50 Order $$96$$ n $$12$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $\GL(2,Z/4)$

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## Group action invariants

 Degree $n$ : $12$ Transitive number $t$ : $50$ Group : $\GL(2,Z/4)$ CHM label : $1/2e[1/16.D(4)^{3}]S(3)$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (3,9)(6,12), (1,7)(3,9)(5,11), (1,2)(3,12)(4,11)(5,10)(6,9)(7,8), (1,5,9)(2,6,10)(3,7,11)(4,8,12) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
8:  $D_{4}$
12:  $D_{6}$
24:  $S_4$, $(C_6\times C_2):C_2$
48:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_3$

## Low degree siblings

12T49 x 2, 12T52, 16T186, 16T193, 24T153, 24T154, 24T155 x 2, 24T156, 24T157, 24T158, 24T159, 24T165, 24T166, 32T392

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 1, 1, 1, 1, 1, 1$ $6$ $2$ $( 4,10)( 5,11)( 6,12)$ $2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $2$ $( 3, 9)( 6,12)$ $2, 2, 2, 1, 1, 1, 1, 1, 1$ $2$ $2$ $( 2, 8)( 4,10)( 6,12)$ $2, 2, 2, 2, 1, 1, 1, 1$ $3$ $2$ $( 2, 8)( 3, 9)( 5,11)( 6,12)$ $4, 2, 2, 2, 2$ $12$ $4$ $( 1, 2)( 3, 6, 9,12)( 4, 5)( 7, 8)(10,11)$ $2, 2, 2, 2, 2, 2$ $12$ $2$ $( 1, 2)( 3, 6)( 4,11)( 5,10)( 7, 8)( 9,12)$ $4, 4, 4$ $12$ $4$ $( 1, 2, 7, 8)( 3, 6, 9,12)( 4, 5,10,11)$ $4, 4, 2, 2$ $12$ $4$ $( 1, 2, 7, 8)( 3, 6)( 4,11,10, 5)( 9,12)$ $6, 6$ $8$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ $6, 3, 3$ $8$ $6$ $( 1, 3, 5)( 2, 4,12, 8,10, 6)( 7, 9,11)$ $3, 3, 3, 3$ $8$ $3$ $( 1, 3,11)( 2, 4, 6)( 5, 7, 9)( 8,10,12)$ $6, 3, 3$ $8$ $6$ $( 1, 3,11, 7, 9, 5)( 2, 4,12)( 6, 8,10)$ $2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$

## Group invariants

 Order: $96=2^{5} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes GAP id: [96, 195]
 Character table:  2 5 4 5 4 5 3 3 3 3 2 2 2 2 5 3 1 . . 1 . . . . . 1 1 1 1 1 1a 2a 2b 2c 2d 4a 2e 4b 4c 6a 6b 3a 6c 2f 2P 1a 1a 1a 1a 1a 2b 1a 2f 2d 3a 3a 3a 3a 1a 3P 1a 2a 2b 2c 2d 4a 2e 4b 4c 2f 2c 1a 2c 2f 5P 1a 2a 2b 2c 2d 4a 2e 4b 4c 6a 6c 3a 6b 2f X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 X.3 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 X.4 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 X.5 2 -2 2 -2 2 . . . . -1 1 -1 1 2 X.6 2 2 2 2 2 . . . . -1 -1 -1 -1 2 X.7 2 . -2 . 2 . . . . -2 . 2 . -2 X.8 2 . -2 . 2 . . . . 1 A -1 -A -2 X.9 2 . -2 . 2 . . . . 1 -A -1 A -2 X.10 3 -1 -1 3 -1 -1 1 1 -1 . . . . 3 X.11 3 -1 -1 3 -1 1 -1 -1 1 . . . . 3 X.12 3 1 -1 -3 -1 -1 -1 1 1 . . . . 3 X.13 3 1 -1 -3 -1 1 1 -1 -1 . . . . 3 X.14 6 . 2 . -2 . . . . . . . . -6 A = -E(3)+E(3)^2 = -Sqrt(-3) = -i3