# Properties

 Label 16T26 Order $$32$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $D_4:C_4$

# Related objects

## Group action invariants

 Degree $n$ : $16$ Transitive number $t$ : $26$ Group : $D_4:C_4$ Parity: $1$ Primitive: No Nilpotency class: $3$ Generators: (1,3,2,4)(5,16,6,15)(7,13,8,14)(9,11,10,12), (1,5)(2,6)(3,4)(7,15)(8,16)(9,14)(10,13)(11,12) $|\Aut(F/K)|$: $4$

## 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, $C_4\times C_2$
16:  $D_{8}$, $QD_{16}$, $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

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

Degree 8: $D_{8}$, $QD_{16}$, $C_2^2:C_4$

## Low degree siblings

16T26, 32T12

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 9]
 Character table:  2 5 3 5 3 3 4 3 4 4 4 4 4 5 5 1a 2a 2b 2c 4a 8a 4b 8b 4c 4d 8c 8d 2d 2e 2P 1a 1a 1a 1a 2b 4c 2b 4c 2d 2d 4c 4c 1a 1a 3P 1a 2a 2b 2c 4b 8c 4a 8d 4c 4d 8a 8b 2d 2e 5P 1a 2a 2b 2c 4a 8d 4b 8c 4c 4d 8b 8a 2d 2e 7P 1a 2a 2b 2c 4b 8b 4a 8a 4c 4d 8d 8c 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 A -A -A A -1 1 A -A 1 -1 X.6 1 -1 -1 1 -A A A -A -1 1 -A A 1 -1 X.7 1 1 -1 -1 A A -A -A -1 1 -A A 1 -1 X.8 1 1 -1 -1 -A -A A A -1 1 A -A 1 -1 X.9 2 . -2 . . . . . 2 -2 . . 2 -2 X.10 2 . 2 . . . . . -2 -2 . . 2 2 X.11 2 . -2 . . B . -B . . B -B -2 2 X.12 2 . -2 . . -B . B . . -B B -2 2 X.13 2 . 2 . . C . C . . -C -C -2 -2 X.14 2 . 2 . . -C . -C . . C C -2 -2 A = -E(4) = -Sqrt(-1) = -i B = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 C = -E(8)+E(8)^3 = -Sqrt(2) = -r2