# Properties

 Label 12T36 Order $$72$$ n $$12$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $S_3\wr C_2$

# Related objects

## Group action invariants

 Degree $n$: $12$ Transitive number $t$: $36$ Group: $S_3\wr C_2$ CHM label: $F_{36}:2(12d)$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,3,5,7,9,11)(2,4,6,8,10,12), (1,9)(2,3)(6,7)(8,12)(10,11) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: $C_3^2:D_4$

## Low degree siblings

6T13 x 2, 9T16, 12T34 x 2, 12T35 x 2, 12T36, 18T34 x 2, 18T36, 24T72 x 2, 36T53, 36T54 x 2

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 1, 1$ $6$ $2$ $( 2, 3)( 4, 8)( 5, 9)( 6, 7)(10,11)$ $3, 3, 1, 1, 1, 1, 1, 1$ $4$ $3$ $( 2, 6,10)( 3, 7,11)$ $6, 2, 2, 1, 1$ $12$ $6$ $( 2, 7,10, 3, 6,11)( 4, 8)( 5, 9)$ $2, 2, 2, 2, 2, 2$ $6$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$ $4, 4, 4$ $18$ $4$ $( 1, 2, 4, 7)( 3, 5, 6,12)( 8,11, 9,10)$ $6, 6$ $12$ $6$ $( 1, 2, 5,10, 9, 6)( 3, 4,11, 8, 7,12)$ $2, 2, 2, 2, 2, 2$ $9$ $2$ $( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)$ $3, 3, 3, 3$ $4$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$

## Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [72, 40]
 Character table: 2 3 2 1 1 2 2 1 3 1 3 2 1 2 1 1 . 1 . 2 1a 2a 3a 6a 2b 4a 6b 2c 3b 2P 1a 1a 3a 3a 1a 2c 3b 1a 3b 3P 1a 2a 1a 2a 2b 4a 2b 2c 1a 5P 1a 2a 3a 6a 2b 4a 6b 2c 3b X.1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 -1 1 -1 1 1 X.3 1 -1 1 -1 1 -1 1 1 1 X.4 1 1 1 1 -1 -1 -1 1 1 X.5 2 . 2 . . . . -2 2 X.6 4 -2 1 1 . . . . -2 X.7 4 . -2 . -2 . 1 . 1 X.8 4 . -2 . 2 . -1 . 1 X.9 4 2 1 -1 . . . . -2