# Properties

 Label 8T17 Order $$32$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_4\wr C_2$

# Related objects

## Group action invariants

 Degree $n$ : $8$ Transitive number $t$ : $17$ Group : $C_4\wr C_2$ CHM label : $[4^{2}]2$ Parity: $-1$ Primitive: No Nilpotency class: $3$ Generators: (1,2,3,8), (1,5)(2,6)(3,7)(4,8) $|\Aut(F/K)|$: $4$

## 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:  $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

## Low degree siblings

8T17, 16T28, 16T42, 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 $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 1, 1, 1, 1$ $2$ $4$ $(4,5,6,7)$ $2, 2, 1, 1, 1, 1$ $2$ $2$ $(4,6)(5,7)$ $4, 1, 1, 1, 1$ $2$ $4$ $(4,7,6,5)$ $4, 4$ $1$ $4$ $(1,2,3,8)(4,5,6,7)$ $4, 2, 2$ $2$ $4$ $(1,2,3,8)(4,6)(5,7)$ $4, 4$ $2$ $4$ $(1,2,3,8)(4,7,6,5)$ $2, 2, 2, 2$ $1$ $2$ $(1,3)(2,8)(4,6)(5,7)$ $4, 2, 2$ $2$ $4$ $(1,3)(2,8)(4,7,6,5)$ $2, 2, 2, 2$ $4$ $2$ $(1,4)(2,5)(3,6)(7,8)$ $8$ $4$ $8$ $(1,4,2,5,3,6,8,7)$ $4, 4$ $4$ $4$ $(1,4,3,6)(2,5,8,7)$ $8$ $4$ $8$ $(1,4,8,7,3,6,2,5)$ $4, 4$ $1$ $4$ $(1,8,3,2)(4,7,6,5)$

## Group invariants

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