# Properties

 Label 12T11 Order $$24$$ n $$12$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $S_3 \times C_4$

# Related objects

## Group action invariants

 Degree $n$ : $12$ Transitive number $t$ : $11$ Group : $S_3 \times C_4$ CHM label : $S(3)[x]C(4)$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12) $|\Aut(F/K)|$: $4$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$
8:  $C_4\times C_2$
12:  $D_{6}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $D_{6}$

## Low degree siblings

12T11, 24T12

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1, 1, 1$ $3$ $2$ $( 2, 6)( 3,11)( 5, 9)( 8,12)$ $12$ $2$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$ $4, 4, 4$ $3$ $4$ $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$ $6, 6$ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ $2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 3)( 2, 8)( 4, 6)( 5,11)( 7, 9)(10,12)$ $4, 4, 4$ $1$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$ $4, 4, 4$ $3$ $4$ $( 1, 4, 7,10)( 2, 9, 8, 3)( 5,12,11, 6)$ $3, 3, 3, 3$ $2$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ $2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ $12$ $2$ $12$ $( 1, 8, 3,10, 5,12, 7, 2, 9, 4,11, 6)$ $4, 4, 4$ $1$ $4$ $( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$

## Group invariants

 Order: $24=2^{3} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes GAP id: [24, 5]
 Character table:  2 3 3 2 3 2 3 3 3 2 3 2 3 3 1 . 1 . 1 . 1 . 1 1 1 1 1a 2a 12a 4a 6a 2b 4b 4c 3a 2c 12b 4d 2P 1a 1a 6a 2c 3a 1a 2c 2c 3a 1a 6a 2c 3P 1a 2a 4b 4c 2c 2b 4d 4a 1a 2c 4d 4b 5P 1a 2a 12a 4a 6a 2b 4b 4c 3a 2c 12b 4d 7P 1a 2a 12b 4c 6a 2b 4d 4a 3a 2c 12a 4b 11P 1a 2a 12b 4c 6a 2b 4d 4a 3a 2c 12a 4b X.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 X.3 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 X.5 1 -1 A -A -1 1 -A A 1 -1 -A A X.6 1 -1 -A A -1 1 A -A 1 -1 A -A X.7 1 1 A A -1 -1 -A -A 1 -1 -A A X.8 1 1 -A -A -1 -1 A A 1 -1 A -A X.9 2 . -1 . -1 . 2 . -1 2 -1 2 X.10 2 . 1 . -1 . -2 . -1 2 1 -2 X.11 2 . A . 1 . B . -1 -2 -A -B X.12 2 . -A . 1 . -B . -1 -2 A B A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i