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Properties

Label 6T13
Order \(72\)
n \(6\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2:D_4$

Related objects

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Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $13$
Group :  $C_3^2:D_4$
CHM label :  $F_{36}(6):2 = [S(3)^{2}]2 = S(3) wr 2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4), (1,4)(2,5)(3,6), (2,4,6)
$|\Aut(F/K)|$:  $1$

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 TypeSizeOrderRepresentative
$ 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 $ $4$ $3$ $(1,3,5)(2,4,6)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 40]
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