# Properties

 Label 16T49 Degree $16$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_8.C_4$

# Related objects

## Group action invariants

 Degree $n$: $16$ Transitive number $t$: $49$ Group: $C_8.C_4$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $8$ Generators: (1,4,6,8,2,3,5,7)(9,12,14,16,10,11,13,15), (1,12,5,16,2,11,6,15)(3,10,8,13,4,9,7,14)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$, $C_4\times C_2$, $Q_8$
$16$:  $C_4:C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8$

## Low degree siblings

32T28

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 15]
 Character table:  2 5 4 5 4 4 4 4 4 5 5 3 3 3 3 1a 2a 2b 8a 8b 8c 8d 4a 4b 4c 8e 8f 8g 8h 2P 1a 1a 1a 4a 4a 4a 4a 2b 2b 2b 4b 4c 4c 4b 3P 1a 2a 2b 8d 8b 8c 8a 4a 4c 4b 8f 8e 8h 8g 5P 1a 2a 2b 8d 8c 8b 8a 4a 4b 4c 8e 8f 8g 8h 7P 1a 2a 2b 8a 8c 8b 8d 4a 4c 4b 8f 8e 8h 8g 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 D -D D -D X.6 1 -1 1 -1 1 1 -1 1 -1 -1 -D D -D D X.7 1 -1 1 1 -1 -1 1 1 -1 -1 D -D -D D X.8 1 -1 1 1 -1 -1 1 1 -1 -1 -D D D -D X.9 2 2 2 . . . . -2 -2 -2 . . . . X.10 2 -2 2 . . . . -2 2 2 . . . . X.11 2 . -2 A B -B -A . C -C . . . . X.12 2 . -2 A -B B -A . -C C . . . . X.13 2 . -2 -A B -B A . -C C . . . . X.14 2 . -2 -A -B B A . C -C . . . . A = -E(8)+E(8)^3 = -Sqrt(2) = -r2 B = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 C = -2*E(4) = -2*Sqrt(-1) = -2i D = -E(4) = -Sqrt(-1) = -i