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Properties

Label 8T45
Degree $8$
Order $576$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(A_4\wr C_2):C_2$

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Show commands: Magma

magma: G := TransitiveGroup(8, 45);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(A_4\wr C_2):C_2$
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), (1,3)(2,8), (1,2,3), (1,5)(2,6)(3,7)(4,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$12$:  $D_{6}$ x 2
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

12T161, 12T163, 12T165 x 2, 16T1032, 16T1034, 18T179, 18T180, 18T185 x 2, 24T1490, 24T1492, 24T1493 x 2, 24T1494 x 2, 24T1495 x 2, 24T1503, 24T1504 x 2, 32T34597 x 2, 32T34598, 36T759, 36T760, 36T762, 36T763, 36T774 x 2, 36T775 x 2, 36T960, 36T961, 36T962 x 2, 36T963 x 2

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 $ $32$ $3$ $(2,3,8)(5,7,6)$
$ 3, 3, 1, 1 $ $32$ $3$ $(2,8,3)(5,7,6)$
$ 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 $ $36$ $4$ $(1,8,2,3)(4,6,5,7)$
$ 2, 2, 2, 2 $ $12$ $2$ $(1,5)(2,6)(3,7)(4,8)$
$ 4, 4 $ $36$ $4$ $(1,4,8,5)(2,7,3,6)$
$ 6, 2 $ $96$ $6$ $(1,5)(2,6,3,7,8,4)$
$ 6, 2 $ $96$ $6$ $(1,5)(2,7,8,4,3,6)$
$ 2, 2, 2, 2 $ $12$ $2$ $(1,5)(2,7)(3,6)(4,8)$
$ 4, 4 $ $36$ $4$ $(1,4,8,5)(2,6,3,7)$

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:  576.8654
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C
Size 1 6 9 12 12 36 16 32 32 36 36 36 72 48 96 96
2 P 1A 1A 1A 1A 1A 1A 3A 3B 3C 2B 2B 2B 2A 3A 3B 3C
3 P 1A 2A 2B 2C 2D 2E 1A 1A 1A 4A 4B 4C 4D 2A 2C 2D
Type
576.8654.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
576.8654.1b R 1 1 1 −1 −1 1 1 1 1 1 −1 −1 1 1 −1 −1
576.8654.1c R 1 1 1 −1 1 −1 1 1 1 −1 1 −1 −1 1 −1 1
576.8654.1d R 1 1 1 1 −1 −1 1 1 1 −1 −1 1 −1 1 1 −1
576.8654.2a R 2 2 2 0 2 0 −1 2 −1 0 2 0 0 −1 0 −1
576.8654.2b R 2 2 2 2 0 0 −1 −1 2 0 0 2 0 −1 −1 0
576.8654.2c R 2 2 2 −2 0 0 −1 −1 2 0 0 −2 0 −1 1 0
576.8654.2d R 2 2 2 0 −2 0 −1 2 −1 0 −2 0 0 −1 0 1
576.8654.4a R 4 4 4 0 0 0 1 −2 −2 0 0 0 0 1 0 0
576.8654.6a R 6 2 −2 0 0 2 3 0 0 −2 0 0 0 −1 0 0
576.8654.6b R 6 2 −2 0 0 −2 3 0 0 2 0 0 0 −1 0 0
576.8654.9a R 9 −3 1 3 3 1 0 0 0 1 −1 −1 −1 0 0 0
576.8654.9b R 9 −3 1 −3 −3 1 0 0 0 1 1 1 −1 0 0 0
576.8654.9c R 9 −3 1 −3 3 −1 0 0 0 −1 −1 1 1 0 0 0
576.8654.9d R 9 −3 1 3 −3 −1 0 0 0 −1 1 −1 1 0 0 0
576.8654.12a R 12 4 −4 0 0 0 −3 0 0 0 0 0 0 1 0 0

magma: CharacterTable(G);