# Properties

 Label 14T34 Order $$1344$$ n $$14$$ Cyclic No Abelian No Solvable No Primitive No $p$-group No

# Related objects

## Group action invariants

 Degree $n$ : $14$ Transitive number $t$ : $34$ CHM label : $2^{3}:L_{7}(14)=[2^{3}]L(7)=[2^{3}]L(3,2)$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,9,11)(2,4,8)(3,13,5)(6,12,10), (2,4)(5,13)(6,12)(9,11), (3,10)(5,12)(6,13)(7,14), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
168:  $\GL(3,2)$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 7: $\GL(3,2)$

## Low degree siblings

8T48 x 2, 14T34, 28T153, 28T159 x 2, 42T210 x 2, 42T211 x 2

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $7$ $2$ $( 3,10)( 5,12)( 6,13)( 7,14)$ $2, 2, 2, 2, 2, 2, 1, 1$ $42$ $2$ $( 1, 8)( 3,12)( 4,11)( 5,10)( 6,14)( 7,13)$ $4, 4, 2, 2, 1, 1$ $84$ $4$ $( 2, 9)( 3, 5,10,12)( 4,11)( 6, 7,13,14)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $42$ $2$ $( 3,12)( 5,10)( 6, 7)(13,14)$ $4, 4, 4, 2$ $168$ $4$ $( 1, 8)( 2, 3,11, 7)( 4,14, 9,10)( 5, 6,12,13)$ $4, 4, 2, 2, 1, 1$ $168$ $4$ $( 2, 3,11,14)( 4, 7, 9,10)( 5, 6)(12,13)$ $3, 3, 3, 3, 1, 1$ $224$ $3$ $( 2, 3,12)( 4, 7, 6)( 5, 9,10)(11,14,13)$ $6, 3, 3, 2$ $224$ $6$ $( 1, 8)( 2, 3, 5)( 4,14,13,11, 7, 6)( 9,10,12)$ $7, 7$ $192$ $7$ $( 1, 2, 3, 4,12, 6, 7)( 5,13,14, 8, 9,10,11)$ $7, 7$ $192$ $7$ $( 1, 9,10, 7, 6, 4, 5)( 2, 3,14,13,11,12, 8)$

## Group invariants

 Order: $1344=2^{6} \cdot 3 \cdot 7$ Cyclic: No Abelian: No Solvable: No GAP id: [1344, 11686]
 Character table:  2 6 6 5 4 5 3 3 1 1 . . 3 1 1 . . . . . 1 1 . . 7 1 . . . . . . . . 1 1 1a 2a 2b 4a 2c 4b 4c 3a 6a 7a 7b 2P 1a 1a 1a 2a 1a 2b 2c 3a 3a 7a 7b 3P 1a 2a 2b 4a 2c 4b 4c 1a 2a 7b 7a 5P 1a 2a 2b 4a 2c 4b 4c 3a 6a 7b 7a 7P 1a 2a 2b 4a 2c 4b 4c 3a 6a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 3 3 -1 -1 -1 1 1 . . A /A X.3 3 3 -1 -1 -1 1 1 . . /A A X.4 6 6 2 2 2 . . . . -1 -1 X.5 7 -1 3 -1 -1 1 -1 1 -1 . . X.6 7 7 -1 -1 -1 -1 -1 1 1 . . X.7 7 -1 -1 -1 3 -1 1 1 -1 . . X.8 8 8 . . . . . -1 -1 1 1 X.9 14 -2 2 -2 2 . . -1 1 . . X.10 21 -3 1 1 -3 -1 1 . . . . X.11 21 -3 -3 1 1 1 -1 . . . . A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7