# Properties

 Label 12T43 Order $$72$$ n $$12$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $S_3\times A_4$

# Related objects

## Group action invariants

 Degree $n$ : $12$ Transitive number $t$ : $43$ Group : $S_3\times A_4$ CHM label : $A(4)[x]S(3)$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,7,10)(2,5,11)(3,6,9), (2,8,11)(3,6,12)(4,7,10), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
12:  $A_4$
18:  $S_3\times C_3$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: $A_4$

Degree 6: None

## Low degree siblings

18T31, 18T32, 24T78, 24T83, 36T21, 36T50, 36T51

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

## Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [72, 44]
 Character table:  2 3 1 3 1 1 1 3 . 2 . 3 2 3 2 1 1 2 2 1 . 2 1 2 1 2 1a 6a 2a 3a 3b 6b 2b 3c 6c 3d 2c 3e 2P 1a 3a 1a 3b 3a 3b 1a 3d 3e 3c 1a 3e 3P 1a 2a 2a 1a 1a 2a 2b 1a 2c 1a 2c 1a 5P 1a 6b 2a 3b 3a 6a 2b 3d 6c 3c 2c 3e 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 A -1 -/A -A /A -1 -/A 1 -A 1 1 X.4 1 /A -1 -A -/A A -1 -A 1 -/A 1 1 X.5 1 -/A 1 -A -/A -A 1 -A 1 -/A 1 1 X.6 1 -A 1 -/A -A -/A 1 -/A 1 -A 1 1 X.7 2 . . 2 2 . . -1 -1 -1 2 -1 X.8 2 . . B /B . . /A -1 A 2 -1 X.9 2 . . /B B . . A -1 /A 2 -1 X.10 3 . -3 . . . 1 . -1 . -1 3 X.11 3 . 3 . . . -1 . -1 . -1 3 X.12 6 . . . . . . . 1 . -2 -3 A = -E(3) = (1-Sqrt(-3))/2 = -b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3