# Properties

 Label 16T48 Order $$32$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_2\times SD_{16}$

# Related objects

## Group action invariants

 Degree $n$ : $16$ Transitive number $t$ : $48$ Group : $C_2\times SD_{16}$ Parity: $1$ Primitive: No Nilpotency class: $3$ Generators: (1,10)(2,9)(3,15)(4,16)(5,6)(7,11)(8,12)(13,14), (1,16,14,12,9,7,6,3)(2,15,13,11,10,8,5,4), (1,3,9,12)(2,4,10,11)(5,15,13,8)(6,16,14,7) $|\Aut(F/K)|$: $4$

## 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:  $QD_{16}$ x 2, $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $QD_{16}$ x 2, $D_4\times C_2$

## Low degree siblings

16T48, 32T27

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 40]
 Character table:  2 5 3 5 3 4 3 4 3 4 4 5 5 4 4 1a 2a 2b 2c 8a 4a 8b 4b 4c 4d 2d 2e 8c 8d 2P 1a 1a 1a 1a 4d 2d 4d 2d 2d 2d 1a 1a 4d 4d 3P 1a 2a 2b 2c 8a 4a 8b 4b 4c 4d 2d 2e 8c 8d 5P 1a 2a 2b 2c 8d 4a 8c 4b 4c 4d 2d 2e 8b 8a 7P 1a 2a 2b 2c 8d 4a 8c 4b 4c 4d 2d 2e 8b 8a X.1 1 1 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 -1 1 -1 X.3 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 X.4 1 -1 1 -1 -1 1 -1 1 1 1 1 1 -1 -1 X.5 1 -1 1 -1 1 -1 1 -1 1 1 1 1 1 1 X.6 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 1 -1 X.7 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 X.8 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 X.9 2 . -2 . . . . . 2 -2 2 -2 . . X.10 2 . 2 . . . . . -2 -2 2 2 . . X.11 2 . -2 . A . -A . . . -2 2 A -A X.12 2 . -2 . -A . A . . . -2 2 -A A X.13 2 . 2 . A . A . . . -2 -2 -A -A X.14 2 . 2 . -A . -A . . . -2 -2 A A A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2