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Properties

Label 16T672
Degree $16$
Order $256$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2.(C_2\times D_4)$

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Show commands: Magma

magma: G := TransitiveGroup(16, 672);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $672$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4^2.(C_2\times D_4)$
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:  (1,12,3,10,2,11,4,9)(5,16,7,14,6,15,8,13), (1,10,6,14,2,9,5,13)(3,12,7,15,4,11,8,16), (1,11,3,14)(2,12,4,13)(5,16,7,10)(6,15,8,9)
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 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3
$32$:  $C_2^2 \wr C_2$
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$
$128$:  $C_2 \wr C_2\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$

Low degree siblings

16T662 x 2, 16T672, 32T3230, 32T3231 x 2, 32T3232, 32T3270, 32T3271, 32T6290 x 2, 32T7435, 32T7447

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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2C $2^{8}$ $8$ $2$ $8$ $( 1,15)( 2,16)( 3, 9)( 4,10)( 5,12)( 6,11)( 7,13)( 8,14)$
2D $2^{8}$ $8$ $2$ $8$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,10)( 6, 9)( 7,12)( 8,11)$
2E $2^{8}$ $8$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,16)(10,15)(11,13)(12,14)$
2F $2^{6},1^{4}$ $16$ $2$ $6$ $( 1, 5)( 2, 6)( 3, 4)( 9,10)(11,16)(12,15)$
2G $2^{8}$ $16$ $2$ $8$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,13)(10,14)(11,16)(12,15)$
4A $4^{4}$ $4$ $4$ $12$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,15,12,16)$
4B $4^{4}$ $8$ $4$ $12$ $( 1,16, 2,15)( 3,10, 4, 9)( 5,11, 6,12)( 7,14, 8,13)$
4C $4^{4}$ $8$ $4$ $12$ $( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,15,10,16)(11,13,12,14)$
4D $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 1, 6, 2, 5)( 3, 8, 4, 7)(11,12)(15,16)$
4E $4^{4}$ $8$ $4$ $12$ $( 1,13, 2,14)( 3,15, 4,16)( 5, 9, 6,10)( 7,11, 8,12)$
4F $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1, 4, 2, 3)( 5, 7, 6, 8)(13,14)(15,16)$
4G $4^{4}$ $32$ $4$ $12$ $( 1,12, 4,14)( 2,11, 3,13)( 5,15, 8,10)( 6,16, 7, 9)$
8A $8^{2}$ $8$ $8$ $14$ $( 1, 8, 5, 4, 2, 7, 6, 3)( 9,15,14,12,10,16,13,11)$
8B $8^{2}$ $8$ $8$ $14$ $( 1, 8, 5, 4, 2, 7, 6, 3)( 9,16,14,11,10,15,13,12)$
8C $8,4,2^{2}$ $16$ $8$ $12$ $( 1, 6, 2, 5)( 3, 4)( 7, 8)( 9,12,13,16,10,11,14,15)$
8D $8^{2}$ $16$ $8$ $14$ $( 1,16, 5,11, 2,15, 6,12)( 3, 9, 8,14, 4,10, 7,13)$
8E $8^{2}$ $16$ $8$ $14$ $( 1,14, 5,10, 2,13, 6, 9)( 3,15, 8,12, 4,16, 7,11)$
8F $8,4,1^{4}$ $16$ $8$ $10$ $( 1, 5, 2, 6)( 9,11,13,15,10,12,14,16)$
8G $8^{2}$ $32$ $8$ $14$ $( 1,10, 4,12, 2, 9, 3,11)( 5,14, 8,16, 6,13, 7,15)$

Malle's constant $a(G)$:     $1/4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $256=2^{8}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $5$
Label:  256.6665
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G
Size 1 1 2 8 8 8 16 16 4 8 8 8 8 16 32 8 8 16 16 16 16 32
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2B 2A 2B 2E 4A 4A 4D 4A 4A 4D 4E
Type
256.6665.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
256.6665.1b R 1 1 1 1 −1 −1 1 −1 1 −1 1 1 −1 −1 1 1 1 −1 −1 −1 −1 1
256.6665.1c R 1 1 1 1 −1 −1 1 1 1 −1 1 1 −1 1 −1 1 1 1 −1 −1 1 −1
256.6665.1d R 1 1 1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 −1 1 −1 1 1 1
256.6665.1e R 1 1 1 1 −1 1 −1 1 1 1 1 1 −1 1 1 −1 −1 −1 −1 1 −1 −1
256.6665.1f R 1 1 1 1 1 −1 −1 −1 1 −1 1 1 1 −1 1 −1 −1 1 1 −1 1 −1
256.6665.1g R 1 1 1 1 1 −1 −1 1 1 −1 1 1 1 1 −1 −1 −1 −1 1 −1 −1 1
256.6665.1h R 1 1 1 1 1 1 1 −1 1 1 1 1 1 −1 −1 1 1 −1 1 1 −1 −1
256.6665.2a R 2 2 2 −2 −2 0 0 0 2 0 −2 2 −2 0 0 0 0 0 2 0 0 0
256.6665.2b R 2 2 2 −2 0 0 −2 0 2 0 2 −2 0 0 0 2 2 0 0 0 0 0
256.6665.2c R 2 2 2 −2 0 0 2 0 2 0 2 −2 0 0 0 −2 −2 0 0 0 0 0
256.6665.2d R 2 2 2 −2 2 0 0 0 2 0 −2 2 2 0 0 0 0 0 −2 0 0 0
256.6665.2e R 2 2 2 2 0 −2 0 0 2 −2 −2 −2 0 0 0 0 0 0 0 2 0 0
256.6665.2f R 2 2 2 2 0 2 0 0 2 2 −2 −2 0 0 0 0 0 0 0 −2 0 0
256.6665.4a R 4 4 4 0 0 0 0 −2 −4 0 0 0 0 2 0 0 0 0 0 0 0 0
256.6665.4b R 4 4 4 0 0 0 0 2 −4 0 0 0 0 −2 0 0 0 0 0 0 0 0
256.6665.4c R 4 4 −4 0 −2 −2 0 0 0 2 0 0 2 0 0 −2 2 0 0 0 0 0
256.6665.4d R 4 4 −4 0 −2 2 0 0 0 −2 0 0 2 0 0 2 −2 0 0 0 0 0
256.6665.4e R 4 4 −4 0 2 −2 0 0 0 2 0 0 −2 0 0 2 −2 0 0 0 0 0
256.6665.4f R 4 4 −4 0 2 2 0 0 0 −2 0 0 −2 0 0 −2 2 0 0 0 0 0
256.6665.8a R 8 −8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −2 0 0 2 0
256.6665.8b R 8 −8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 −2 0

magma: CharacterTable(G);