# Properties

 Label 8T18 Order $$32$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $V_4 \wr C_2$

# Learn more about

## Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $18$
Group :  $V_4 \wr C_2$
CHM label :  $E(8):E_{4}=[2^{2}]D(4)$
Parity:  $1$
Primitive:  No
Generators:  (1,3)(2,8)(4,5)(6,7), (1,5,2,6)(3,7,8,4), (1,8)(2,3)
$|\Aut(F/K)|$:  $4$
Low degree resolvents:
 2: $C_2$ x 7 4: $V_4$ x 7 8: $D_4$ x 6, $C_2^3$ 16: $D_4\times C_2$ x 3

## Subfields

Degree 2: $C_2$

Degree 4: $D_4$ x 3

## Low degree siblings

8T18 x 7, 16T39 x 6, 16T46, 32T24
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1$ $2$ $2$ $(4,5)(6,7)$ $2, 2, 1, 1, 1, 1$ $2$ $2$ $(4,6)(5,7)$ $2, 2, 1, 1, 1, 1$ $2$ $2$ $(4,7)(5,6)$ $2, 2, 2, 2$ $2$ $2$ $(1,2)(3,8)(4,5)(6,7)$ $2, 2, 2, 2$ $2$ $2$ $(1,2)(3,8)(4,6)(5,7)$ $2, 2, 2, 2$ $1$ $2$ $(1,2)(3,8)(4,7)(5,6)$ $2, 2, 2, 2$ $2$ $2$ $(1,3)(2,8)(4,5)(6,7)$ $2, 2, 2, 2$ $1$ $2$ $(1,3)(2,8)(4,6)(5,7)$ $2, 2, 2, 2$ $4$ $2$ $(1,4)(2,7)(3,6)(5,8)$ $4, 4$ $4$ $4$ $(1,4,2,7)(3,6,8,5)$ $4, 4$ $4$ $4$ $(1,4,3,6)(2,7,8,5)$ $4, 4$ $4$ $4$ $(1,4,8,5)(2,7,3,6)$ $2, 2, 2, 2$ $1$ $2$ $(1,8)(2,3)(4,5)(6,7)$

## Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 27]
 Character table:  2 5 4 4 4 4 4 5 4 5 3 3 3 3 5 1a 2a 2b 2c 2d 2e 2f 2g 2h 2i 4a 4b 4c 2j 2P 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 2f 2h 2j 1a 3P 1a 2a 2b 2c 2d 2e 2f 2g 2h 2i 4a 4b 4c 2j 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 -1 1 -1 1 X.6 1 1 -1 -1 -1 1 1 -1 1 -1 1 1 -1 1 X.7 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 1 X.8 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 X.9 2 . . 2 . . 2 -2 -2 . . . . -2 X.10 2 . . -2 . . 2 2 -2 . . . . -2 X.11 2 -2 . . . 2 -2 . -2 . . . . 2 X.12 2 . -2 . 2 . -2 . 2 . . . . -2 X.13 2 . 2 . -2 . -2 . 2 . . . . -2 X.14 2 2 . . . -2 -2 . -2 . . . . 2