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Properties

Label 16T15
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2 \times (C_8:C_2)$

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Show commands: Magma

magma: G := TransitiveGroup(16, 15);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $15$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2 \times (C_8:C_2)$
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:  (1,8,5,12,10,16,14,4)(2,7,6,11,9,15,13,3), (1,2)(3,12)(4,11)(5,6)(7,16)(8,15)(9,10)(13,14), (1,5,10,14)(2,6,9,13)(3,15,11,7)(4,16,12,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $C_4\times C_2$ x 6, $C_2^3$
$16$:  $C_8:C_2$ x 2, $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$, $C_8:C_2$ x 2

Low degree siblings

16T15, 32T1

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 $ $2$ $2$ $( 3,11)( 4,12)( 7,15)( 8,16)$
$ 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, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3,12)( 4,11)( 5, 6)( 7,16)( 8,15)( 9,10)(13,14)$
$ 8, 8 $ $2$ $8$ $( 1, 3, 5, 7,10,11,14,15)( 2, 4, 6, 8, 9,12,13,16)$
$ 8, 8 $ $2$ $8$ $( 1, 3,14,15,10,11, 5, 7)( 2, 4,13,16, 9,12, 6, 8)$
$ 8, 8 $ $2$ $8$ $( 1, 4, 5, 8,10,12,14,16)( 2, 3, 6, 7, 9,11,13,15)$
$ 8, 8 $ $2$ $8$ $( 1, 4,14,16,10,12, 5, 8)( 2, 3,13,15, 9,11, 6, 7)$
$ 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 $ $2$ $4$ $( 1, 5,10,14)( 2, 6, 9,13)( 3,15,11, 7)( 4,16,12, 8)$
$ 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 $ $2$ $4$ $( 1, 6,10,13)( 2, 5, 9,14)( 3,16,11, 8)( 4,15,12, 7)$
$ 8, 8 $ $2$ $8$ $( 1, 7, 5,11,10,15,14, 3)( 2, 8, 6,12, 9,16,13, 4)$
$ 8, 8 $ $2$ $8$ $( 1, 7,14, 3,10,15, 5,11)( 2, 8,13, 4, 9,16, 6,12)$
$ 8, 8 $ $2$ $8$ $( 1, 8, 5,12,10,16,14, 4)( 2, 7, 6,11, 9,15,13, 3)$
$ 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:  32.37
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A1 4A-1 4B1 4B-1 4C 4D 8A1 8A-1 8B1 8B-1 8C1 8C-1 8D1 8D-1
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 4A-1 4A1 4A-1 4A1 4A-1 4A1 4A-1 4A1
Type
32.37.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.37.1b R 1 1 −1 −1 −1 1 1 1 −1 −1 1 −1 −1 −1 1 1 −1 −1 1 1
32.37.1c R 1 1 −1 −1 −1 1 1 1 −1 −1 1 −1 1 1 −1 −1 1 1 −1 −1
32.37.1d R 1 1 −1 −1 1 −1 1 1 −1 −1 −1 1 −1 −1 −1 −1 1 1 1 1
32.37.1e R 1 1 −1 −1 1 −1 1 1 −1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
32.37.1f R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1
32.37.1g R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
32.37.1h R 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
32.37.1i1 C 1 1 −1 −1 −1 1 −1 −1 1 1 −1 1 −i i −i i −i i −i i
32.37.1i2 C 1 1 −1 −1 −1 1 −1 −1 1 1 −1 1 i −i i −i i −i i −i
32.37.1j1 C 1 1 −1 −1 1 −1 −1 −1 1 1 1 −1 −i i i −i i −i −i i
32.37.1j2 C 1 1 −1 −1 1 −1 −1 −1 1 1 1 −1 i −i −i i −i i i −i
32.37.1k1 C 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 −i i −i i i −i i −i
32.37.1k2 C 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 i −i i −i −i i −i i
32.37.1l1 C 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −i i i −i −i i i −i
32.37.1l2 C 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 i −i −i i i −i −i i
32.37.2a1 C 2 −2 −2 2 0 0 −2i 2i −2i 2i 0 0 0 0 0 0 0 0 0 0
32.37.2a2 C 2 −2 −2 2 0 0 2i −2i 2i −2i 0 0 0 0 0 0 0 0 0 0
32.37.2b1 C 2 −2 2 −2 0 0 −2i 2i 2i −2i 0 0 0 0 0 0 0 0 0 0
32.37.2b2 C 2 −2 2 −2 0 0 2i −2i −2i 2i 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);