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Properties

Label 8T38
Degree $8$
Order $192$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\wr A_4$

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Show commands: Magma

magma: G := TransitiveGroup(8, 38);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $38$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\wr A_4$
CHM label:   $[2^{4}]A(4)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8)(2,3)(4,5)(6,7), (1,2,3)(5,6,7), (4,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$
$96$:  $C_2^4:C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Low degree siblings

8T38, 16T425, 16T427, 24T288 x 2, 24T425 x 2, 32T2185 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
$1^{8}$ $1$ $1$ $()$
$2,1^{6}$ $4$ $2$ $(4,8)$
$2^{2},1^{4}$ $6$ $2$ $(3,7)(4,8)$
$3^{2},1^{2}$ $16$ $3$ $(2,3,4)(6,7,8)$
$6,1^{2}$ $16$ $6$ $(2,3,4,6,7,8)$
$3^{2},1^{2}$ $16$ $3$ $(2,4,3)(6,8,7)$
$6,1^{2}$ $16$ $6$ $(2,4,7,6,8,3)$
$2^{3},1^{2}$ $4$ $2$ $(2,6)(3,7)(4,8)$
$2^{4}$ $12$ $2$ $(1,2)(3,4)(5,6)(7,8)$
$4,2^{2}$ $24$ $4$ $(1,2)(3,4,7,8)(5,6)$
$3^{2},2$ $16$ $6$ $(1,2,3)(4,8)(5,6,7)$
$6,2$ $16$ $6$ $(1,2,3,5,6,7)(4,8)$
$3^{2},2$ $16$ $6$ $(1,2,4)(3,7)(5,6,8)$
$6,2$ $16$ $6$ $(1,2,4,5,6,8)(3,7)$
$4^{2}$ $12$ $4$ $(1,2,5,6)(3,4,7,8)$
$2^{4}$ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $192=2^{6} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  192.201
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 3A1 3A-1 4A 4B 6A1 6A-1 6B1 6B-1 6C1 6C-1
Size 1 1 4 4 6 12 16 16 12 24 16 16 16 16 16 16
2 P 1A 1A 1A 1A 1A 1A 3A-1 3A1 2A 2D 3A-1 3A1 3A1 3A-1 3A-1 3A1
3 P 1A 2A 2B 2C 2D 2E 1A 1A 4A 4B 2C 2C 2A 2A 2B 2B
Type
192.201.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
192.201.1b R 1 1 −1 −1 1 1 1 1 1 −1 1 1 −1 −1 −1 −1
192.201.1c1 C 1 1 1 1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
192.201.1c2 C 1 1 1 1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
192.201.1d1 C 1 1 −1 −1 1 1 ζ3−1 ζ3 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
192.201.1d2 C 1 1 −1 −1 1 1 ζ3 ζ3−1 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1
192.201.3a R 3 3 3 3 3 −1 0 0 −1 −1 0 0 0 0 0 0
192.201.3b R 3 3 −3 −3 3 −1 0 0 −1 1 0 0 0 0 0 0
192.201.4a R 4 −4 −2 2 0 0 1 1 0 0 −1 −1 1 1 −1 −1
192.201.4b R 4 −4 2 −2 0 0 1 1 0 0 −1 −1 −1 −1 1 1
192.201.4c1 C 4 −4 −2 2 0 0 ζ3−1 ζ3 0 0 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3
192.201.4c2 C 4 −4 −2 2 0 0 ζ3 ζ3−1 0 0 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1
192.201.4d1 C 4 −4 2 −2 0 0 ζ3−1 ζ3 0 0 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3
192.201.4d2 C 4 −4 2 −2 0 0 ζ3 ζ3−1 0 0 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1
192.201.6a R 6 6 0 0 −2 −2 0 0 2 0 0 0 0 0 0 0
192.201.6b R 6 6 0 0 −2 2 0 0 −2 0 0 0 0 0 0 0

magma: CharacterTable(G);