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Properties

Label 16T30
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2:C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 30);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $30$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4^2:C_2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14,16,12)(2,13,15,11)(3,8,5,10)(4,7,6,9), (1,5)(2,6)(3,16)(4,15)(7,14)(8,13)(9,12)(10,11), (1,4,2,3)(5,16,6,15)(7,13,8,14)(9,11,10,12)
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
$8$:  $D_{4}$ x 2, $C_2^3$
$16$:  $D_4\times C_2$, $Q_8:C_2$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4\times C_2$, $Q_8:C_2$ x 2

Low degree siblings

16T30, 32T16

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{16}$ $1$ $1$ $0$ $()$
2A $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2B $2^{8}$ $1$ $2$ $8$ $( 1,15)( 2,16)( 3, 6)( 4, 5)( 7,10)( 8, 9)(11,14)(12,13)$
2C $2^{8}$ $1$ $2$ $8$ $( 1,16)( 2,15)( 3, 5)( 4, 6)( 7, 9)( 8,10)(11,13)(12,14)$
2D $2^{6},1^{4}$ $4$ $2$ $6$ $( 3, 4)( 5, 6)( 7, 9)( 8,10)(11,14)(12,13)$
2E $2^{8}$ $4$ $2$ $8$ $( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,13)( 8,14)( 9,11)(10,12)$
4A $4^{4}$ $2$ $4$ $12$ $( 1, 4, 2, 3)( 5,16, 6,15)( 7,13, 8,14)( 9,11,10,12)$
4B $4^{4}$ $2$ $4$ $12$ $( 1, 9,15, 8)( 2,10,16, 7)( 3,12, 6,13)( 4,11, 5,14)$
4C1 $4^{4}$ $2$ $4$ $12$ $( 1,14,16,12)( 2,13,15,11)( 3, 8, 5,10)( 4, 7, 6, 9)$
4C-1 $4^{4}$ $2$ $4$ $12$ $( 1, 5, 2, 6)( 3,15, 4,16)( 7,12, 8,11)( 9,14,10,13)$
4D1 $4^{4}$ $2$ $4$ $12$ $( 1, 8,15, 9)( 2, 7,16,10)( 3,13, 6,12)( 4,14, 5,11)$
4D-1 $4^{4}$ $2$ $4$ $12$ $( 1,12,16,14)( 2,11,15,13)( 3,10, 5, 8)( 4, 9, 6, 7)$
4E $4^{4}$ $4$ $4$ $12$ $( 1,12, 2,11)( 3, 9, 4,10)( 5, 7, 6, 8)(13,16,14,15)$
4F $4^{4}$ $4$ $4$ $12$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,13, 6,14)( 7,15, 8,16)$

Malle's constant $a(G)$:     $1/6$

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:  32.31
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A 4B 4C1 4C-1 4D1 4D-1 4E 4F
Size 1 1 1 1 4 4 2 2 2 2 2 2 4 4
2 P 1A 1A 1A 1A 1A 1A 2A 2B 2C 2A 2B 2C 2A 2A
Type
32.31.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.31.1b R 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
32.31.1c R 1 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1
32.31.1d R 1 1 1 1 −1 1 −1 −1 −1 −1 1 1 −1 1
32.31.1e R 1 1 1 1 −1 1 −1 −1 1 1 −1 −1 1 −1
32.31.1f R 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 1 −1
32.31.1g R 1 1 1 1 1 −1 −1 −1 1 1 −1 −1 −1 1
32.31.1h R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
32.31.2a R 2 −2 −2 2 0 0 −2 2 0 0 0 0 0 0
32.31.2b R 2 −2 −2 2 0 0 2 −2 0 0 0 0 0 0
32.31.2c1 C 2 −2 2 −2 0 0 0 0 −2i 2i 0 0 0 0
32.31.2c2 C 2 −2 2 −2 0 0 0 0 2i −2i 0 0 0 0
32.31.2d1 C 2 2 −2 −2 0 0 0 0 0 0 −2i 2i 0 0
32.31.2d2 C 2 2 −2 −2 0 0 0 0 0 0 2i −2i 0 0

magma: CharacterTable(G);