# Properties

 Label 16T42 Order $$32$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_4\wr C_2$

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## Group action invariants

 Degree $n$ : $16$ Transitive number $t$ : $42$ Group : $C_4\wr C_2$ Parity: $1$ Primitive: No Nilpotency class: $3$ Generators: (1,3,5,15)(2,4,6,16)(7,8)(9,10)(11,12)(13,14), (1,10)(2,9)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15) $|\Aut(F/K)|$: $8$

## 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:  $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$, $C_4\wr C_2$ x 2

## Low degree siblings

8T17 x 2, 16T28, 32T14

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

## Group invariants

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