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Properties

Label 24T2691
24T2691 1 8 1->8 14 1->14 2 7 2->7 13 2->13 3 6 3->6 16 3->16 4 5 4->5 15 4->15 10 5->10 19 5->19 9 6->9 20 6->20 11 7->11 18 7->18 12 8->12 17 8->17 9->1 21 9->21 10->2 22 10->22 11->3 23 11->23 12->4 24 12->24 13->5 13->15 14->6 14->16 15->8 15->14 16->7 16->13 17->10 17->20 18->9 18->19 19->11 19->17 20->12 20->18 21->4 21->23 22->3 22->24 23->1 23->22 24->2 24->21
Degree $24$
Order $1152$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4\wr C_2\times C_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(24, 2691);
 

Group invariants

Abstract group:  $A_4\wr C_2\times C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $1152=2^{7} \cdot 3^{2}$
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 $n$:  $24$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2691$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,8,12,4,5,10,2,7,11,3,6,9)(13,15,14,16)(17,20,18,19)(21,23,22,24)$, $(1,14,6,20,12,24,2,13,5,19,11,23)(3,16,7,18,9,21,4,15,8,17,10,22)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$8$:  $C_4\times C_2$
$12$:  $D_{6}$, $C_{12}$ x 2, $C_6\times C_2$
$18$:  $S_3\times C_3$
$24$:  $S_3 \times C_4$, 24T2
$36$:  $C_6\times S_3$
$72$:  24T65
$288$:  $A_4\wr C_2$
$576$:  12T158

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $S_3\times C_3$

Degree 8: None

Degree 12: $C_6\times S_3$

Low degree siblings

32T96703, 36T1634 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Conjugacy classes not computed

Copy content magma:ConjugacyClasses(G);
 

Character table

56 x 56 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed