# Properties

 Label 6T13 Degree $6$ Order $72$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3^2:D_4$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(6, 13);

## Group action invariants

 Degree $n$: $6$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $13$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_3^2:D_4$ CHM label: $F_{36}(6):2 = [S(3)^{2}]2 = S(3) wr 2$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (2,4), (1,4)(2,5)(3,6), (2,4,6) magma: Generators(G);

## 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

## Low degree siblings

6T13, 9T16, 12T34 x 2, 12T35 x 2, 12T36 x 2, 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 $1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 1, 1, 1, 1$ $6$ $2$ $(4,6)$ $2, 2, 1, 1$ $9$ $2$ $(3,5)(4,6)$ $3, 1, 1, 1$ $4$ $3$ $(2,4,6)$ $3, 2, 1$ $12$ $6$ $(2,4,6)(3,5)$ $2, 2, 2$ $6$ $2$ $(1,2)(3,4)(5,6)$ $4, 2$ $18$ $4$ $(1,2)(3,4,5,6)$ $6$ $12$ $6$ $(1,2,3,4,5,6)$ $3, 3$ $4$ $3$ $(1,3,5)(2,4,6)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 72.40 magma: IdentifyGroup(G);
 Character table:  2 3 2 3 1 1 2 2 1 1 3 2 1 . 2 1 1 . 1 2 1a 2a 2b 3a 6a 2c 4a 6b 3b 2P 1a 1a 1a 3a 3a 1a 2b 3b 3b 3P 1a 2a 2b 1a 2a 2c 4a 2c 1a 5P 1a 2a 2b 3a 6a 2c 4a 6b 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 

magma: CharacterTable(G);