Properties

 Label 8T18 Order $$32$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_2^2 \wr C_2$

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Group action invariants

 Degree $n$ : $8$ Transitive number $t$ : $18$ Group : $C_2^2 \wr C_2$ CHM label : $E(8):E_{4}=[2^{2}]D(4)$ Parity: $1$ Primitive: No Nilpotency class: $2$ Generators: (1,3)(2,8)(4,6)(5,7), (4,5)(6,7), (4,6)(5,7), (1,8)(2,3)(4,5)(6,7), (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 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$ x 3

Low degree siblings

8T18 x 7, 16T39 x 6, 16T46, 32T24

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

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