# Properties

 Label 16T30 Order $$32$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_4.D_4$

# Related objects

## Group action invariants

 Degree $n$ : $16$ Transitive number $t$ : $30$ Group : $C_4.D_4$ Parity: $1$ Primitive: No Nilpotency class: $2$ Generators: (1,14,16,12)(2,13,15,11)(3,8,5,10)(4,7,6,9), (1,5)(2,6)(3,16)(4,15)(7,14)(8,13)(9,12)(10,11), (1,4,2,3)(5,16,6,15)(7,13,8,14)(9,11,10,12) $|\Aut(F/K)|$: $4$

## 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:  $D_4\times C_2$, $Q_8:C_2$ x 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: $D_4\times C_2$, $Q_8:C_2$ x 2

## Low degree siblings

16T30, 32T16

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 31]
 Character table:  2 5 3 5 3 4 4 3 4 4 4 3 4 5 5 1a 2a 2b 2c 4a 4b 4c 4d 4e 4f 4g 4h 2d 2e 2P 1a 1a 1a 1a 2b 2b 2b 2d 2d 2e 2b 2e 1a 1a 3P 1a 2a 2b 2c 4a 4b 4c 4e 4d 4h 4g 4f 2d 2e 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 X.12 2 . -2 . . . . . . -A . A 2 -2 X.13 2 . -2 . . . . A -A . . . -2 2 X.14 2 . -2 . . . . -A A . . . -2 2 A = -2*E(4) = -2*Sqrt(-1) = -2i