# Properties

 Label 8T16 Order $$32$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $(C_8:C_2):C_2$

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## Group action invariants

 Degree $n$ : $8$ Transitive number $t$ : $16$ Group : $(C_8:C_2):C_2$ CHM label : $1/2[2^{4}]4$ Parity: $-1$ Primitive: No Nilpotency class: $3$ Generators: (1,2,3,4,5,6,7,8), (2,6)(3,7) $|\Aut(F/K)|$: $2$

## 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: $C_4$

## Low degree siblings

8T16, 16T36, 16T41 x 2, 32T22

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 $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1$ $4$ $2$ $(3,7)(4,8)$ $2, 2, 1, 1, 1, 1$ $2$ $2$ $(2,6)(4,8)$ $8$ $4$ $8$ $(1,2,3,4,5,6,7,8)$ $8$ $4$ $8$ $(1,2,3,8,5,6,7,4)$ $2, 2, 2, 2$ $4$ $2$ $(1,3)(2,4)(5,7)(6,8)$ $4, 4$ $2$ $4$ $(1,3,5,7)(2,4,6,8)$ $4, 4$ $2$ $4$ $(1,3,5,7)(2,8,6,4)$ $8$ $4$ $8$ $(1,4,7,6,5,8,3,2)$ $8$ $4$ $8$ $(1,4,3,6,5,8,7,2)$ $2, 2, 2, 2$ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

## Group invariants

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