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Properties

Label 16T25
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^2 \times D_4$

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Show commands: Magma

magma: G := TransitiveGroup(16, 25);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $25$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2 \times D_4$
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:  (1,9)(2,10)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15), (1,16)(2,15)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14), (7,10)(8,9)(11,14)(12,13), (1,5)(2,6)(3,16)(4,15)(7,12)(8,11)(9,14)(10,13)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $D_{4}$ x 4, $C_2^3$ x 15
$16$:  $D_4\times C_2$ x 6, $C_2^4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4\times C_2$ x 6

Low degree siblings

16T25 x 7, 32T11

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

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:  32.46
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D
Size 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2B 2B 2B 2B
Type
32.46.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.46.1b R 1 −1 1 −1 1 −1 1 −1 −1 −1 −1 1 1 −1 1 1 −1 1 1 −1
32.46.1c R 1 −1 1 −1 1 −1 1 −1 −1 1 −1 −1 1 1 −1 1 1 −1 −1 1
32.46.1d R 1 −1 1 −1 1 −1 1 −1 1 −1 1 1 −1 −1 1 −1 1 −1 −1 1
32.46.1e R 1 −1 1 −1 1 −1 1 −1 1 1 1 −1 −1 1 −1 −1 −1 1 1 −1
32.46.1f R 1 −1 1 1 −1 −1 −1 1 −1 −1 1 −1 −1 1 1 1 −1 −1 1 1
32.46.1g R 1 −1 1 1 −1 −1 −1 1 −1 1 1 1 −1 −1 −1 1 1 1 −1 −1
32.46.1h R 1 −1 1 1 −1 −1 −1 1 1 −1 −1 −1 1 1 1 −1 1 1 −1 −1
32.46.1i R 1 −1 1 1 −1 −1 −1 1 1 1 −1 1 1 −1 −1 −1 −1 −1 1 1
32.46.1j R 1 1 1 −1 −1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 1 −1 1 −1
32.46.1k R 1 1 1 −1 −1 1 −1 −1 −1 1 1 −1 1 −1 1 −1 −1 1 −1 1
32.46.1l R 1 1 1 −1 −1 1 −1 −1 1 −1 −1 1 −1 1 −1 1 −1 1 −1 1
32.46.1m R 1 1 1 −1 −1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 −1 1 −1
32.46.1n R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
32.46.1o R 1 1 1 1 1 1 1 1 −1 1 −1 1 −1 1 1 −1 −1 −1 −1 −1
32.46.1p R 1 1 1 1 1 1 1 1 1 −1 1 −1 1 −1 −1 1 −1 −1 −1 −1
32.46.2a R 2 −2 −2 −2 2 2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
32.46.2b R 2 −2 −2 2 −2 2 2 −2 0 0 0 0 0 0 0 0 0 0 0 0
32.46.2c R 2 2 −2 −2 −2 −2 2 2 0 0 0 0 0 0 0 0 0 0 0 0
32.46.2d R 2 2 −2 2 2 −2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);