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Properties

Label 8T46
Degree $8$
Order $576$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4^2:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(8, 46);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $46$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4^2:C_4$
CHM label:   $1/2[S(4)^{2}]2$
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:  (1,8)(4,5), (1,5)(2,7,3,6)(4,8), (1,3)(2,8), (1,2,3)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$36$:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

12T160, 12T162, 16T1030, 16T1031, 18T182, 18T184, 24T1489, 24T1491, 24T1505, 24T1506 x 2, 24T1508, 32T34594, 36T764, 36T765, 36T766, 36T767, 36T964, 36T965

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1 $ $6$ $2$ $(4,5)(6,7)$
$ 2, 2, 2, 2 $ $9$ $2$ $(1,8)(2,3)(4,5)(6,7)$
$ 3, 1, 1, 1, 1, 1 $ $16$ $3$ $(2,3,8)$
$ 3, 2, 2, 1 $ $48$ $6$ $(2,3,8)(4,5)(6,7)$
$ 3, 3, 1, 1 $ $64$ $3$ $(2,3,8)(5,6,7)$
$ 2, 2, 1, 1, 1, 1 $ $36$ $2$ $(3,8)(6,7)$
$ 4, 2, 1, 1 $ $72$ $4$ $(1,8,2,3)(6,7)$
$ 4, 4 $ $36$ $4$ $(1,8,2,3)(4,7,5,6)$
$ 4, 2, 2 $ $72$ $4$ $(1,5)(2,7,3,6)(4,8)$
$ 8 $ $72$ $8$ $(1,5,3,6,8,4,2,7)$
$ 4, 2, 2 $ $72$ $4$ $(1,5)(2,6)(3,7,8,4)$
$ 8 $ $72$ $8$ $(1,4,3,6,2,7,8,5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  576.8652
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 4A 4B 4C1 4C-1 6A 8A1 8A-1
Size 1 6 9 36 16 64 36 72 72 72 48 72 72
2 P 1A 1A 1A 1A 3A 3B 2B 2A 2C 2C 3A 4A 4A
3 P 1A 2A 2B 2C 1A 1A 4A 4B 4C-1 4C1 2A 8A-1 8A1
Type
576.8652.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
576.8652.1b R 1 1 1 1 1 1 1 1 −1 −1 1 −1 −1
576.8652.1c1 C 1 1 1 −1 1 1 −1 −1 −i i 1 −i i
576.8652.1c2 C 1 1 1 −1 1 1 −1 −1 i −i 1 i −i
576.8652.4a R 4 4 4 0 −2 1 0 0 0 0 −2 0 0
576.8652.4b R 4 4 4 0 1 −2 0 0 0 0 1 0 0
576.8652.6a R 6 2 −2 2 3 0 −2 0 0 0 −1 0 0
576.8652.6b R 6 2 −2 −2 3 0 2 0 0 0 −1 0 0
576.8652.9a R 9 −3 1 1 0 0 1 −1 1 1 0 −1 −1
576.8652.9b R 9 −3 1 1 0 0 1 −1 −1 −1 0 1 1
576.8652.9c1 C 9 −3 1 −1 0 0 −1 1 −i i 0 i −i
576.8652.9c2 C 9 −3 1 −1 0 0 −1 1 i −i 0 −i i
576.8652.12a R 12 4 −4 0 −3 0 0 0 0 0 1 0 0

magma: CharacterTable(G);