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Properties

Label 16T375
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2.(C_2\times C_4)$

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Show commands: Magma

magma: G := TransitiveGroup(16, 375);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $375$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4^2.(C_2\times C_4)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14,2,13)(3,12,7,16)(4,11,8,15)(5,10)(6,9), (1,4,2,3)(5,7,6,8)(9,16)(10,15)(11,13)(12,14)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$32$:  $C_2^3 : C_4 $
$64$:  16T154

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $C_2^3: C_4$

Low degree siblings

16T375, 32T850 x 2, 32T851, 32T1824 x 2, 32T2008

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $2^{4},1^{8}$ $2$ $2$ $4$ $(1,2)(3,4)(5,6)(7,8)$
2C $2^{6},1^{4}$ $8$ $2$ $6$ $( 1, 6)( 2, 5)( 7, 8)(11,16)(12,15)(13,14)$
2D $2^{6},1^{4}$ $8$ $2$ $6$ $( 1, 2)( 3, 7)( 4, 8)(11,15)(12,16)(13,14)$
4A $4^{4}$ $4$ $4$ $12$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,14,10,13)(11,15,12,16)$
4B $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 1, 5, 2, 6)( 3, 8, 4, 7)(11,12)(15,16)$
4C $4^{3},2^{2}$ $16$ $4$ $11$ $( 1,16, 2,15)( 3,13, 8,10)( 4,14, 7, 9)( 5,11)( 6,12)$
4D1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,16)(10,15)(11,13)(12,14)$
4D-1 $4^{3},2^{2}$ $16$ $4$ $11$ $( 1,13, 2,14)( 3,16, 7,12)( 4,15, 8,11)( 5,10)( 6, 9)$
4E1 $4^{3},2^{2}$ $16$ $4$ $11$ $( 1,12, 6,16)( 2,11, 5,15)( 3,14)( 4,13)( 7, 9, 8,10)$
4E-1 $4^{3},2^{2}$ $16$ $4$ $11$ $( 1,14, 2,13)( 3,12, 7,16)( 4,11, 8,15)( 5,10)( 6, 9)$
8A1 $8^{2}$ $8$ $8$ $14$ $( 1, 8, 5, 3, 2, 7, 6, 4)( 9,12,13,15,10,11,14,16)$
8A-1 $8^{2}$ $8$ $8$ $14$ $( 1, 7, 5, 4, 2, 8, 6, 3)( 9,12,13,15,10,11,14,16)$

Malle's constant $a(G)$:     $1/4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $128=2^{7}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $5$
Label:  128.144
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 4A 4B 4C 4D1 4D-1 4E1 4E-1 8A1 8A-1
Size 1 1 2 8 8 4 8 16 16 16 16 16 8 8
2 P 1A 1A 1A 1A 1A 2A 2B 2C 2B 2D 2C 2D 4A 4A
Type
128.144.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
128.144.1b R 1 1 1 1 1 1 1 −1 −1 −1 1 1 −1 −1
128.144.1c R 1 1 1 1 1 1 1 −1 1 1 −1 −1 −1 −1
128.144.1d R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1
128.144.1e1 C 1 1 1 −1 −1 1 1 −1 −i i i −i 1 1
128.144.1e2 C 1 1 1 −1 −1 1 1 −1 i −i −i i 1 1
128.144.1f1 C 1 1 1 −1 −1 1 1 1 −i i −i i −1 −1
128.144.1f2 C 1 1 1 −1 −1 1 1 1 i −i i −i −1 −1
128.144.2a R 2 2 2 −2 2 2 −2 0 0 0 0 0 0 0
128.144.2b R 2 2 2 2 −2 2 −2 0 0 0 0 0 0 0
128.144.4a R 4 4 4 0 0 −4 0 0 0 0 0 0 0 0
128.144.4b1 C 4 4 −4 0 0 0 0 0 0 0 0 0 −2i 2i
128.144.4b2 C 4 4 −4 0 0 0 0 0 0 0 0 0 2i −2i
128.144.8a R 8 −8 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);