# Properties

 Label 16T10 Order $$16$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_2^2 : C_4$

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## Group action invariants

 Degree $n$ : $16$ Transitive number $t$ : $10$ Group : $C_2^2 : C_4$ Parity: $1$ Primitive: No Nilpotency class: $2$ Generators: (1,9)(2,10)(3,4)(5,13)(6,14)(7,8)(11,12)(15,16), (1,3,6,15)(2,4,5,16)(7,9,11,14)(8,10,12,13) $|\Aut(F/K)|$: $16$

## Low degree resolvents

|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, $C_4\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 4

Degree 8: $C_4\times C_2$, $D_4$ x 2, $C_2^2:C_4$ x 2

## Low degree siblings

8T10 x 2

Siblings are shown with degree $\leq 47$

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

## Conjugacy Classes

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

## Group invariants

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