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Properties

Label 16T24
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^2 : C_8$

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Show commands: Magma

magma: G := TransitiveGroup(16, 24);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $24$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2 : C_8$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3,5,7,10,11,14,15)(2,4,6,8,9,12,13,16), (1,10)(2,9)(3,4)(5,14)(6,13)(7,8)(11,12)(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, $C_8$ x 2, $C_4\times C_2$
$16$:  $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_8$, $C_8:C_2$, $C_2^2:C_4$

Low degree siblings

16T24, 32T10

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, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3,12)( 4,11)( 7,16)( 8,15)$
$ 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, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3,11)( 4,12)( 5, 6)( 7,15)( 8,16)( 9,10)(13,14)$
$ 8, 8 $ $2$ $8$ $( 1, 3, 5, 7,10,11,14,15)( 2, 4, 6, 8, 9,12,13,16)$
$ 8, 8 $ $2$ $8$ $( 1, 3,13,16,10,11, 6, 8)( 2, 4,14,15, 9,12, 5, 7)$
$ 8, 8 $ $2$ $8$ $( 1, 4, 5, 8,10,12,14,16)( 2, 3, 6, 7, 9,11,13,15)$
$ 8, 8 $ $2$ $8$ $( 1, 4,13,15,10,12, 6, 7)( 2, 3,14,16, 9,11, 5, 8)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 5,10,14)( 2, 6, 9,13)( 3, 7,11,15)( 4, 8,12,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 5,10,14)( 2, 6, 9,13)( 3,16,11, 8)( 4,15,12, 7)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 6,10,13)( 2, 5, 9,14)( 3, 8,11,16)( 4, 7,12,15)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 6,10,13)( 2, 5, 9,14)( 3,15,11, 7)( 4,16,12, 8)$
$ 8, 8 $ $2$ $8$ $( 1, 7, 6,12,10,15,13, 4)( 2, 8, 5,11, 9,16,14, 3)$
$ 8, 8 $ $2$ $8$ $( 1, 7,14, 3,10,15, 5,11)( 2, 8,13, 4, 9,16, 6,12)$
$ 8, 8 $ $2$ $8$ $( 1, 8, 6,11,10,16,13, 3)( 2, 7, 5,12, 9,15,14, 4)$
$ 8, 8 $ $2$ $8$ $( 1, 8,14, 4,10,16, 5,12)( 2, 7,13, 3, 9,15, 6,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,16)( 8,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,14)( 6,13)( 7,15)( 8,16)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1,13,10, 6)( 2,14, 9, 5)( 3,16,11, 8)( 4,15,12, 7)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1,14,10, 5)( 2,13, 9, 6)( 3,15,11, 7)( 4,16,12, 8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
Label:  32.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A1 4A-1 4B1 4B-1 4C1 4C-1 8A1 8A-1 8A3 8A-3 8B1 8B-1 8B3 8B-3
Size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 4A1 4A-1 4A-1 4B-1 4B1 4B-1 4A1 4B1
Type
32.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.5.1b R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1
32.5.1c R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
32.5.1d R 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
32.5.1e1 C 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 −i i i −i −i i i −i
32.5.1e2 C 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 i −i −i i i −i −i i
32.5.1f1 C 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −i i i −i i −i −i i
32.5.1f2 C 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 i −i −i i −i i i −i
32.5.1g1 C 1 −1 1 −1 −1 1 −ζ82 ζ82 ζ82 −ζ82 −ζ82 ζ82 ζ83 −ζ8 ζ8 −ζ83 ζ8 −ζ83 ζ83 −ζ8
32.5.1g2 C 1 −1 1 −1 −1 1 ζ82 −ζ82 −ζ82 ζ82 ζ82 −ζ82 −ζ8 ζ83 −ζ83 ζ8 −ζ83 ζ8 −ζ8 ζ83
32.5.1g3 C 1 −1 1 −1 −1 1 −ζ82 ζ82 ζ82 −ζ82 −ζ82 ζ82 −ζ83 ζ8 −ζ8 ζ83 −ζ8 ζ83 −ζ83 ζ8
32.5.1g4 C 1 −1 1 −1 −1 1 ζ82 −ζ82 −ζ82 ζ82 ζ82 −ζ82 ζ8 −ζ83 ζ83 −ζ8 ζ83 −ζ8 ζ8 −ζ83
32.5.1h1 C 1 −1 1 −1 1 −1 −ζ82 ζ82 ζ82 −ζ82 ζ82 −ζ82 ζ83 −ζ8 ζ8 −ζ83 −ζ8 ζ83 −ζ83 ζ8
32.5.1h2 C 1 −1 1 −1 1 −1 ζ82 −ζ82 −ζ82 ζ82 −ζ82 ζ82 −ζ8 ζ83 −ζ83 ζ8 ζ83 −ζ8 ζ8 −ζ83
32.5.1h3 C 1 −1 1 −1 1 −1 −ζ82 ζ82 ζ82 −ζ82 ζ82 −ζ82 −ζ83 ζ8 −ζ8 ζ83 ζ8 −ζ83 ζ83 −ζ8
32.5.1h4 C 1 −1 1 −1 1 −1 ζ82 −ζ82 −ζ82 ζ82 −ζ82 ζ82 ζ8 −ζ83 ζ83 −ζ8 −ζ83 ζ8 −ζ8 ζ83
32.5.2a R 2 2 −2 −2 0 0 −2 −2 2 2 0 0 0 0 0 0 0 0 0 0
32.5.2b R 2 2 −2 −2 0 0 2 2 −2 −2 0 0 0 0 0 0 0 0 0 0
32.5.2c1 C 2 −2 −2 2 0 0 −2i 2i −2i 2i 0 0 0 0 0 0 0 0 0 0
32.5.2c2 C 2 −2 −2 2 0 0 2i −2i 2i −2i 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);