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Properties

Label 16T18
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2 \times (C_4\times C_2):C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 18);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $18$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2 \times (C_4\times C_2):C_2$
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:  (1,16)(2,15)(3,6)(4,5)(7,10)(8,9)(11,14)(12,13), (1,5,10,13)(2,6,9,14)(3,8,11,15)(4,7,12,16), (1,9)(2,10)(3,4)(5,14)(6,13)(7,8)(11,12)(15,16), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)
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
$8$:  $C_2^3$ x 15
$16$:  $Q_8:C_2$ x 2, $C_2^4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$, $Q_8:C_2$ x 2

Low degree siblings

16T18 x 5, 32T4

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:  32.48
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A1 4A-1 4B1 4B-1 4C 4D 4E 4F 4G 4H
Size 1 1 1 1 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A
Type
32.48.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.48.1b R 1 1 −1 −1 −1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1 −1 1 1
32.48.1c R 1 1 −1 −1 −1 −1 1 1 1 −1 1 1 −1 −1 1 1 −1 −1 −1 1
32.48.1d R 1 1 −1 −1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 1 −1 1 1 −1
32.48.1e R 1 1 −1 −1 −1 1 1 −1 1 −1 −1 −1 1 1 1 −1 1 1 −1 −1
32.48.1f R 1 1 −1 −1 1 −1 −1 1 −1 1 1 1 −1 −1 1 −1 1 1 −1 −1
32.48.1g R 1 1 −1 −1 1 −1 1 1 −1 −1 −1 −1 1 1 −1 1 −1 1 1 −1
32.48.1h R 1 1 −1 −1 1 1 −1 −1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 1
32.48.1i R 1 1 −1 −1 1 1 1 −1 −1 −1 1 1 −1 −1 −1 −1 1 −1 1 1
32.48.1j R 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1 1 1
32.48.1k R 1 1 1 1 −1 −1 1 −1 −1 1 1 1 1 1 −1 −1 −1 1 −1 1
32.48.1l R 1 1 1 1 −1 1 −1 1 −1 −1 1 1 1 1 1 −1 −1 −1 1 −1
32.48.1m R 1 1 1 1 −1 1 1 1 −1 1 −1 −1 −1 −1 −1 1 1 −1 −1 −1
32.48.1n R 1 1 1 1 1 −1 −1 −1 1 −1 1 1 1 1 −1 1 1 −1 −1 −1
32.48.1o R 1 1 1 1 1 −1 1 −1 1 1 −1 −1 −1 −1 1 −1 −1 −1 1 −1
32.48.1p R 1 1 1 1 1 1 −1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 −1 1
32.48.2a1 C 2 −2 −2 2 0 0 0 0 0 0 −2i 2i −2i 2i 0 0 0 0 0 0
32.48.2a2 C 2 −2 −2 2 0 0 0 0 0 0 2i −2i 2i −2i 0 0 0 0 0 0
32.48.2b1 C 2 −2 2 −2 0 0 0 0 0 0 −2i 2i 2i −2i 0 0 0 0 0 0
32.48.2b2 C 2 −2 2 −2 0 0 0 0 0 0 2i −2i −2i 2i 0 0 0 0 0 0

magma: CharacterTable(G);