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Properties

Label 8T42
8T42 1 2 1->2 3 1->3 5 1->5 2->3 6 2->6 8 2->8 3->1 7 3->7 4 4->8
Degree $8$
Order $288$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4\wr C_2$

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Copy content magma:G := TransitiveGroup(8, 42);
 

Group invariants

Abstract group:  $A_4\wr C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $288=2^{5} \cdot 3^{2}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $8$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $42$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[A(4)^{2}]2$
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,3)(2,8)$, $(1,2,3)$, $(1,5)(2,6)(3,7)(4,8)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

12T126, 12T128, 12T129, 16T708, 18T112, 18T113, 24T692, 24T694, 24T695, 24T702, 24T703, 24T704, 32T9306, 36T316, 36T318, 36T456, 36T457, 36T458, 36T459

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{8}$ $1$ $1$ $0$ $()$
2A $2^{2},1^{4}$ $6$ $2$ $2$ $(1,3)(2,8)$
2B $2^{4}$ $9$ $2$ $4$ $(1,8)(2,3)(4,5)(6,7)$
2C $2^{4}$ $12$ $2$ $4$ $(1,5)(2,6)(3,7)(4,8)$
3A1 $3,1^{5}$ $8$ $3$ $2$ $(4,7,5)$
3A-1 $3,1^{5}$ $8$ $3$ $2$ $(4,5,7)$
3B1 $3^{2},1^{2}$ $16$ $3$ $4$ $(1,8,2)(4,6,5)$
3B-1 $3^{2},1^{2}$ $16$ $3$ $4$ $(1,2,8)(4,5,6)$
3C $3^{2},1^{2}$ $32$ $3$ $4$ $(1,2,3)(4,5,6)$
4A $4^{2}$ $36$ $4$ $6$ $(1,6,8,7)(2,5,3,4)$
6A1 $3,2^{2},1$ $24$ $6$ $4$ $(1,3)(2,8)(4,5,7)$
6A-1 $3,2^{2},1$ $24$ $6$ $4$ $(1,3)(2,8)(4,7,5)$
6B1 $6,2$ $48$ $6$ $6$ $(1,6,8,5,2,4)(3,7)$
6B-1 $6,2$ $48$ $6$ $6$ $(1,4,2,5,8,6)(3,7)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A1 3A-1 3B1 3B-1 3C 4A 6A1 6A-1 6B1 6B-1
Size 1 6 9 12 8 8 16 16 32 36 24 24 48 48
2 P 1A 1A 1A 1A 3A-1 3A1 3B-1 3B1 3C 2B 3A1 3A-1 3B1 3B-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 4A 2A 2A 2C 2C
Type
288.1025.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
288.1025.1b R 1 1 1 −1 1 1 1 1 1 −1 1 1 −1 −1
288.1025.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1
288.1025.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3
288.1025.1d1 C 1 1 1 −1 ζ3−1 ζ3 ζ3−1 ζ3 1 −1 ζ3 ζ3−1 −ζ3 −ζ3−1
288.1025.1d2 C 1 1 1 −1 ζ3 ζ3−1 ζ3 ζ3−1 1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3
288.1025.2a R 2 2 2 0 −1 −1 2 2 −1 0 −1 −1 0 0
288.1025.2b1 C 2 2 2 0 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −1 0 −ζ3−1 −ζ3 0 0
288.1025.2b2 C 2 2 2 0 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −1 0 −ζ3 −ζ3−1 0 0
288.1025.6a R 6 2 −2 0 3 3 0 0 0 0 −1 −1 0 0
288.1025.6b1 C 6 2 −2 0 3ζ3−1 3ζ3 0 0 0 0 −ζ3 −ζ3−1 0 0
288.1025.6b2 C 6 2 −2 0 3ζ3 3ζ3−1 0 0 0 0 −ζ3−1 −ζ3 0 0
288.1025.9a R 9 −3 1 3 0 0 0 0 0 −1 0 0 0 0
288.1025.9b R 9 −3 1 −3 0 0 0 0 0 1 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $9 x^{8} + 24 x^{7} + 16 x^{6} + 6 t x^{4} + 8 t x^{3} + \left(t^{2} - 12 t + 48\right)$ Copy content Toggle raw display