# Properties

 Label 8T30 Order $$64$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $(((C_4 \times C_2): C_2):C_2):C_2$

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## Group action invariants

 Degree $n$ : $8$ Transitive number $t$ : $30$ Group : $(((C_4 \times C_2): C_2):C_2):C_2$ CHM label : $1/2[2^{4}]cD(4)$ Parity: $-1$ Primitive: No Generators: (1,2,3,8)(4,5,6,7), (2,6)(3,7), (1,3)(4,8)(5,7) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $V_4$
8:  $D_{4}$ x 2, $C_4\times C_2$
16:  $C_2^2:C_4$
32:  $C_2^3 : C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

## Low degree siblings

8T30 x 3, 16T143 x 2, 16T167 x 2, 16T168 x 2, 16T169 x 2, 32T157 x 2, 32T177, 32T178

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

## Group invariants

 Order: $64=2^{6}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [64, 34]
 Character table:  2 6 4 4 3 5 3 3 3 3 4 4 4 6 1a 2a 4a 2b 2c 4b 4c 4d 4e 2d 4f 4g 2e 2P 1a 1a 2c 1a 1a 2a 2a 2d 2d 1a 2e 2c 1a 3P 1a 2a 4a 2b 2c 4c 4b 4e 4d 2d 4f 4g 2e X.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 X.3 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 X.5 1 -1 -1 1 1 A -A -A A -1 1 -1 1 X.6 1 -1 -1 1 1 -A A A -A -1 1 -1 1 X.7 1 -1 1 -1 1 A -A A -A -1 1 1 1 X.8 1 -1 1 -1 1 -A A -A A -1 1 1 1 X.9 2 -2 . . 2 . . . . 2 -2 . 2 X.10 2 2 . . 2 . . . . -2 -2 . 2 X.11 4 . . . -4 . . . . . . . 4 X.12 4 . -2 . . . . . . . . 2 -4 X.13 4 . 2 . . . . . . . . -2 -4 A = -E(4) = -Sqrt(-1) = -i