# Properties

 Label 16T1036 Order $672$ n $16$ Cyclic No Abelian No Solvable No Primitive No $p$-group No

# Related objects

## Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $1036$
CHM label :  $t16n1036$
Parity:  $-1$
Primitive:  No
Generators:   ( 1, 4,13, 5, 7,16,11, 9, 2, 3,14, 6, 8,15,12,10), ( 1, 8,16, 5,11, 4,10)( 2, 7,15, 6,12, 3, 9)
$|\textrm{Aut}(F/K)|$:  $2$
Low degree resolvents:
 2: 2T1 336: 8T43

## Subfields

Degree 2: None

Degree 4: None

Degree 8: $\PGL(2,7)$

## Low degree siblings

16T1036b
A number field with this Galois group has exactly one arithmetically equivalent field.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ $2, 2, 2, 2, 2, 2, 2, 1, 1$ $56$ $2$ $( 3, 4)( 5, 8)( 6, 7)( 9,14)(10,13)(11,15)(12,16)$ $6, 6, 2, 1, 1$ $56$ $6$ $( 3, 4)( 5, 9,15, 8,14,11)( 6,10,16, 7,13,12)$ $6, 6, 2, 1, 1$ $56$ $6$ $( 1, 2)( 5,10,15, 7,14,12)( 6, 9,16, 8,13,11)$ $3, 3, 3, 3, 1, 1, 1, 1$ $56$ $3$ $( 5,14,15)( 6,13,16)( 7,10,12)( 8, 9,11)$ $6, 6, 2, 2$ $56$ $6$ $( 1, 2)( 3, 4)( 5,13,15, 6,14,16)( 7, 9,12, 8,10,11)$ $7, 7, 1, 1$ $48$ $7$ $( 3, 6,13,11,16, 9, 8)( 4, 5,14,12,15,10, 7)$ $14, 2$ $48$ $14$ $( 1, 2)( 3, 5,13,12,16,10, 8, 4, 6,14,11,15, 9, 7)$ $4, 4, 4, 4$ $42$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,16,10,15)(11,13,12,14)$ $16$ $42$ $16$ $( 1, 4, 6,11, 8,14,16, 9, 2, 3, 5,12, 7,13,15,10)$ $16$ $42$ $16$ $( 1, 3, 6,12, 8,13,16,10, 2, 4, 5,11, 7,14,15, 9)$ $8, 8$ $42$ $8$ $( 1, 4, 5,14, 2, 3, 6,13)( 7,11,10,15, 8,12, 9,16)$ $8, 8$ $42$ $8$ $( 1, 3, 5,13, 2, 4, 6,14)( 7,12,10,16, 8,11, 9,15)$ $16$ $42$ $16$ $( 1, 4, 6,15,12,13, 9, 8, 2, 3, 5,16,11,14,10, 7)$ $16$ $42$ $16$ $( 1, 3, 6,16,12,14, 9, 7, 2, 4, 5,15,11,13,10, 8)$

## Group invariants

 Order: $672=2^{5} \cdot 3 \cdot 7$ Cyclic: No Abelian: No Solvable: No
 Character table: ``` 2 5 5 2 2 2 2 2 1 1 4 4 4 4 4 4 4 3 1 1 1 1 1 1 1 . . . . . . . . . 7 1 1 . . . . . 1 1 . . . . . . . 1a 2a 2b 6a 6b 3a 6c 7a 14a 4a 16a 16b 8a 8b 16c 16d 2P 1a 1a 1a 3a 3a 3a 3a 7a 7a 2a 8a 8a 4a 4a 8b 8b 3P 1a 2a 2b 2b 2b 1a 2a 7a 14a 4a 16d 16c 8b 8a 16a 16b 5P 1a 2a 2b 6b 6a 3a 6c 7a 14a 4a 16d 16c 8b 8a 16a 16b 7P 1a 2a 2b 6a 6b 3a 6c 1a 2a 4a 16a 16b 8a 8b 16c 16d 11P 1a 2a 2b 6b 6a 3a 6c 7a 14a 4a 16c 16d 8b 8a 16b 16a 13P 1a 2a 2b 6a 6b 3a 6c 7a 14a 4a 16c 16d 8b 8a 16b 16a X.1 1 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 -1 -1 X.3 6 6 . . . . . -1 -1 -2 . . 2 2 . . X.4 6 -6 . . . . . -1 1 . B -B D -D -C C X.5 6 -6 . . . . . -1 1 . C -C -D D B -B X.6 6 -6 . . . . . -1 1 . -C C -D D -B B X.7 6 -6 . . . . . -1 1 . -B B D -D C -C X.8 6 6 . . . . . -1 -1 2 D D . . -D -D X.9 6 6 . . . . . -1 -1 2 -D -D . . D D X.10 7 7 -1 -1 -1 1 1 . . -1 1 1 -1 -1 1 1 X.11 7 7 1 1 1 1 1 . . -1 -1 -1 -1 -1 -1 -1 X.12 8 8 -2 1 1 -1 -1 1 1 . . . . . . . X.13 8 8 2 -1 -1 -1 -1 1 1 . . . . . . . X.14 8 -8 . . . 2 -2 1 -1 . . . . . . . X.15 8 -8 . A -A -1 1 1 -1 . . . . . . . X.16 8 -8 . -A A -1 1 1 -1 . . . . . . . A = -E(3)+E(3)^2 = -Sqrt(-3) = -i3 B = -E(16)-E(16)^7 C = -E(16)^3-E(16)^5 D = -E(8)+E(8)^3 = -Sqrt(2) = -r2 ```