Properties

 Label 16T7 Order $$16$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $D_8$

Related objects

Group action invariants

 Degree $n$ : $16$ Transitive number $t$ : $7$ Group : $D_8$ Parity: $1$ Primitive: No Nilpotency class: $2$ Generators: (1,14,2,13)(3,8,4,7)(5,10,6,9)(11,16,12,15), (1,15,2,16)(3,5,4,6)(7,10,8,9)(11,14,12,13), (1,6)(2,5)(3,15)(4,16)(7,12)(8,11)(9,14)(10,13) $|\Aut(F/K)|$: $16$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $C_2^3$, $Q_8$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

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

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ $4, 4, 4, 4$ $2$ $4$ $( 1, 3, 2, 4)( 5,16, 6,15)( 7,13, 8,14)( 9,12,10,11)$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,11)( 8,12)( 9,13)(10,14)$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,12)( 8,11)( 9,14)(10,13)$ $4, 4, 4, 4$ $2$ $4$ $( 1, 7, 2, 8)( 3,14, 4,13)( 5,11, 6,12)( 9,16,10,15)$ $4, 4, 4, 4$ $2$ $4$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,13, 6,14)( 7,15, 8,16)$ $4, 4, 4, 4$ $2$ $4$ $( 1,11, 2,12)( 3,10, 4, 9)( 5, 7, 6, 8)(13,16,14,15)$ $4, 4, 4, 4$ $2$ $4$ $( 1,13, 2,14)( 3, 7, 4, 8)( 5, 9, 6,10)(11,15,12,16)$ $4, 4, 4, 4$ $2$ $4$ $( 1,15, 2,16)( 3, 5, 4, 6)( 7,10, 8, 9)(11,14,12,13)$

Group invariants

 Order: $16=2^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [16, 12]
 Character table:  2 4 4 3 4 4 3 3 3 3 3 1a 2a 4a 2b 2c 4b 4c 4d 4e 4f 2P 1a 1a 2a 1a 1a 2a 2a 2a 2a 2a 3P 1a 2a 4a 2b 2c 4b 4c 4d 4e 4f X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 -1 -1 -1 1 1 1 X.3 1 1 -1 -1 -1 1 1 -1 -1 1 X.4 1 1 -1 1 1 -1 1 -1 1 -1 X.5 1 1 -1 1 1 1 -1 1 -1 -1 X.6 1 1 1 -1 -1 -1 1 1 -1 -1 X.7 1 1 1 -1 -1 1 -1 -1 1 -1 X.8 1 1 1 1 1 -1 -1 -1 -1 1 X.9 2 -2 . 2 -2 . . . . . X.10 2 -2 . -2 2 . . . . .