# Properties

 Label 16T45 Order $$32$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_8:C_2^2$

# Learn more about

## Group action invariants

 Degree $n$ : $16$ Transitive number $t$ : $45$ Group : $C_8:C_2^2$ Parity: $1$ Primitive: No Nilpotency class: $3$ Generators: (1,9)(2,10)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15), (1,4,2,3)(5,16,6,15)(7,11,8,12)(9,14,10,13), (1,11)(2,12)(3,9)(4,10)(5,8)(6,7)(13,15)(14,16) $|\Aut(F/K)|$: $4$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4\times C_2$

## Low degree siblings

8T15 x 2, 16T35, 16T38 x 2, 32T21

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 43]
 Character table: 2 5 3 5 3 4 4 3 3 3 3 4 1a 2a 2b 4a 4b 2c 8a 2d 8b 2e 4c 2P 1a 1a 1a 2b 2b 1a 4c 1a 4c 1a 2b 3P 1a 2a 2b 4a 4b 2c 8a 2d 8b 2e 4c 5P 1a 2a 2b 4a 4b 2c 8a 2d 8b 2e 4c 7P 1a 2a 2b 4a 4b 2c 8a 2d 8b 2e 4c X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 1 -1 1 -1 1 1 X.3 1 -1 1 -1 1 1 1 -1 1 -1 1 X.4 1 -1 1 1 -1 -1 -1 1 1 -1 1 X.5 1 -1 1 1 -1 -1 1 -1 -1 1 1 X.6 1 1 1 -1 -1 -1 -1 -1 1 1 1 X.7 1 1 1 -1 -1 -1 1 1 -1 -1 1 X.8 1 1 1 1 1 1 -1 -1 -1 -1 1 X.9 2 . 2 . -2 2 . . . . -2 X.10 2 . 2 . 2 -2 . . . . -2 X.11 4 . -4 . . . . . . . .