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Properties

Label 16T1497
Degree $16$
Order $2304$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_4^2:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(16, 1497);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1497$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_4^2:C_4$
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:  (1,9,5,8,4,14,16,12)(2,10,6,7,3,13,15,11), (1,2)(3,6)(4,5)(7,12,9,8,11,10)(13,14)(15,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$72$:  $C_3^2:D_4$
$144$:  12T79
$1152$:  $S_4\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $S_4\wr C_2$

Low degree siblings

12T237 x 2, 12T238 x 2, 16T1496 x 2, 16T1497, 24T5093 x 2, 24T5117 x 2, 24T5118 x 2, 24T5119, 24T5120 x 2, 24T5121 x 2, 24T5122 x 2, 24T5123 x 2, 24T5124, 24T5125 x 2, 24T5126 x 2, 24T5127 x 2, 24T5128 x 2, 32T205436 x 2, 32T205437 x 2, 32T205438, 32T205439, 36T3213 x 2, 36T3215 x 2, 36T3216 x 2, 36T3218, 36T3224, 36T3449 x 2, 36T3450 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, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,11)( 8,12)( 9,14)(10,13)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 7,14)( 8,13)( 9,11)(10,12)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $64$ $3$ $( 3,16, 5)( 4,15, 6)( 9,14,11)(10,13,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $3$ $( 9,11,14)(10,12,13)$
$ 3, 3, 2, 2, 2, 2, 1, 1 $ $48$ $6$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,11, 9)( 8,12,10)$
$ 4, 4, 4, 4 $ $36$ $4$ $( 1, 5, 4,16)( 2, 6, 3,15)( 7,13,11,10)( 8,14,12, 9)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $36$ $2$ $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7, 8)( 9,13)(10,14)(11,12)$
$ 4, 4, 2, 2, 2, 2 $ $72$ $4$ $( 1, 5, 4,16)( 2, 6, 3,15)( 7, 8)( 9,13)(10,14)(11,12)$
$ 8, 8 $ $144$ $8$ $( 1, 9, 5, 8, 4,14,16,12)( 2,10, 6, 7, 3,13,15,11)$
$ 4, 4, 4, 4 $ $144$ $4$ $( 1,11, 2,12)( 3, 8, 4, 7)( 5,13,15, 9)( 6,14,16,10)$
$ 8, 8 $ $144$ $8$ $( 1, 8,16, 9, 4,12, 5,14)( 2, 7,15,10, 3,11, 6,13)$
$ 4, 4, 4, 4 $ $144$ $4$ $( 1,13, 2,14)( 3, 9, 4,10)( 5, 7,15,12)( 6, 8,16,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $36$ $2$ $( 5,16)( 6,15)(11,14)(12,13)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $72$ $4$ $( 1, 4)( 2, 3)( 7,11, 9,14)( 8,12,10,13)$
$ 4, 4, 4, 4 $ $36$ $4$ $( 1, 6, 4,15)( 2, 5, 3,16)( 7,14, 9,11)( 8,13,10,12)$
$ 6, 2, 2, 2, 2, 2 $ $48$ $6$ $( 1, 5)( 2, 6)( 3,15)( 4,16)( 7,13, 9, 8,14,10)(11,12)$
$ 6, 2, 2, 2, 2, 2 $ $16$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,13,11,10,14,12)(15,16)$
$ 6, 6, 2, 2 $ $64$ $6$ $( 1, 3, 6, 2, 4, 5)( 7,12, 9, 8,11,10)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 5)( 2, 6)( 3,15)( 4,16)( 7,10)( 8, 9)(11,13)(12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,12)( 8,11)( 9,13)(10,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $(11,14)(12,13)$
$ 4, 4, 2, 2, 2, 2 $ $36$ $4$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,11, 9,14)( 8,12,10,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $36$ $2$ $( 1,15)( 2,16)( 3, 5)( 4, 6)( 7, 9)( 8,10)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $4$ $( 7,14, 9,11)( 8,13,10,12)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1 $ $96$ $6$ $( 3,16, 5)( 4,15, 6)( 9,14)(10,13)$
$ 4, 4, 3, 3, 1, 1 $ $96$ $12$ $( 1, 6,15)( 2, 5,16)( 7,14,11, 9)( 8,13,12,10)$
$ 6, 4, 4, 2 $ $96$ $12$ $( 1, 5, 4,16)( 2, 6, 3,15)( 7,13, 9, 8,14,10)(11,12)$
$ 6, 2, 2, 2, 2, 2 $ $96$ $6$ $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7, 8)( 9,13,11,10,14,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $36$ $2$ $( 1,16)( 2,15)( 3, 4)( 5, 6)( 7,10)( 8, 9)(11,13)(12,14)$
$ 4, 4, 2, 2, 2, 2 $ $36$ $4$ $( 1, 5,15, 3)( 2, 6,16, 4)( 7,13)( 8,14)( 9,12)(10,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 6)( 4, 5)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $12$ $4$ $( 1, 5,15, 3)( 2, 6,16, 4)( 7, 8)( 9,10)(11,12)(13,14)$
$ 12, 4 $ $192$ $12$ $( 1, 9, 5, 8, 4,14, 2,10, 6, 7, 3,13)(11,16,12,15)$
$ 4, 4, 4, 4 $ $72$ $4$ $( 1,11,16,13)( 2,12,15,14)( 3,10, 6, 7)( 4, 9, 5, 8)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1, 7, 2, 8)( 3,13, 4,14)( 5,12, 6,11)( 9,16,10,15)$
$ 12, 4 $ $192$ $12$ $( 1, 8,16, 9, 4,12, 2, 7,15,10, 3,11)( 5,14, 6,13)$
$ 4, 4, 4, 4 $ $72$ $4$ $( 1, 8, 5,11)( 2, 7, 6,12)( 3, 9,15,13)( 4,10,16,14)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1,10, 2, 9)( 3, 7, 4, 8)( 5,14, 6,13)(11,15,12,16)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $2304=2^{8} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  2304.ee
magma: IdentifyGroup(G);
 
Character table:    40 x 40 character table

magma: CharacterTable(G);