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Properties

Label 24T2476
24T2476 1 9 1->9 21 1->21 23 1->23 2 10 2->10 22 2->22 24 2->24 3 4 3->4 12 3->12 16 3->16 11 4->11 15 4->15 5 6 5->6 13 5->13 18 5->18 14 6->14 17 6->17 7 7->10 7->12 7->15 8 8->9 8->11 8->16 9->13 9->17 10->14 10->18 11->12 20 11->20 19 12->19 13->14 13->22 14->21 15->16 15->24 16->23 17->1 17->18 18->2 19->6 19->20 19->23 20->5 20->24 21->3 21->22 22->4 23->8 24->7
Degree $24$
Order $768$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^3:\GL(2,\mathbb{Z}/4)$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(24, 2476);
 

Group invariants

Abstract group:  $C_2^3:\GL(2,\mathbb{Z}/4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $768=2^{8} \cdot 3$
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 $t$:  $2476$
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:  $(1,23)(2,24)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$, $(1,21)(2,22)(3,16)(4,15)(5,18)(6,17)(7,12)(8,11)(9,13)(10,14)(19,23)(20,24)$, $(1,9,17)(2,10,18)(3,12,19,6,14,21)(4,11,20,5,13,22)(7,15,24)(8,16,23)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $D_{4}$ x 2, $C_2^3$
$12$:  $D_{6}$ x 3
$16$:  $D_4\times C_2$
$24$:  $S_4$ x 3, $S_3 \times C_2^2$, $(C_6\times C_2):C_2$ x 2
$48$:  $S_4\times C_2$ x 9, 24T25
$96$:  $V_4^2:S_3$, 12T48 x 3, 12T49 x 6
$192$:  12T100 x 3, 24T398 x 3
$384$:  12T135 x 2, 12T139

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $S_3$

Degree 8: None

Degree 12: $(C_6\times C_2):C_2$, 12T106, 12T135

Low degree siblings

24T1932 x 12, 24T1974 x 12, 24T2204 x 24, 24T2469 x 24, 24T2470 x 48, 24T2471 x 8, 24T2472 x 12, 24T2475 x 4, 24T2476 x 7, 24T2498 x 48, 24T2525 x 24, 32T34970 x 4

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