# Properties

 Label 14T22 Order $$588$$ n $$14$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No

# Related objects

## Group action invariants

 Degree $n$ : $14$ Transitive number $t$ : $22$ CHM label : $[1/6_-.F_{42}(7)^{2}]2_{2}$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (2,4,6,8,10,12,14), (1,6,13,8)(2,9,12,5)(3,4,11,10)(7,14), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8), (1,11,9)(2,4,8)(3,5,13)(6,12,10) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
12:  $C_3 : C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 7: None

## Low degree siblings

28T75, 42T119, 42T125

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $7, 1, 1, 1, 1, 1, 1, 1$ $12$ $7$ $( 2, 4, 6, 8,10,12,14)$ $7, 7$ $12$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$ $7, 7$ $12$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$ $7, 7$ $12$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$ $2, 2, 2, 2, 2, 2, 1, 1$ $49$ $2$ $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$ $3, 3, 3, 3, 1, 1$ $98$ $3$ $( 3, 5, 9)( 4,10, 6)( 7,13,11)( 8,12,14)$ $6, 6, 1, 1$ $98$ $6$ $( 3,11, 9,13, 5, 7)( 4, 8, 6,14,10,12)$ $4, 4, 4, 2$ $147$ $4$ $( 1, 6,13, 8)( 2, 9,12, 5)( 3, 4,11,10)( 7,14)$ $4, 4, 4, 2$ $147$ $4$ $( 1,12,11, 8)( 2, 7, 4, 5)( 3,14, 9, 6)(10,13)$

## Group invariants

 Order: $588=2^{2} \cdot 3 \cdot 7^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [588, 33]
 Character table:  2 2 . . . . 2 1 1 2 2 3 1 . . . . 1 1 1 . . 7 2 2 2 2 2 . . . . . 1a 7a 7b 7c 7d 2a 3a 6a 4a 4b 2P 1a 7a 7c 7d 7b 1a 3a 3a 2a 2a 3P 1a 7a 7d 7b 7c 2a 1a 2a 4b 4a 5P 1a 7a 7c 7d 7b 2a 3a 6a 4a 4b 7P 1a 1a 1a 1a 1a 2a 3a 6a 4b 4a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 1 1 1 -1 -1 X.3 1 1 1 1 1 -1 1 -1 D -D X.4 1 1 1 1 1 -1 1 -1 -D D X.5 2 2 2 2 2 -2 -1 1 . . X.6 2 2 2 2 2 2 -1 -1 . . X.7 12 5 -2 -2 -2 . . . . . X.8 12 -2 A C B . . . . . X.9 12 -2 B A C . . . . . X.10 12 -2 C B A . . . . . A = E(7)^2-2*E(7)^3-2*E(7)^4+E(7)^5 B = E(7)-2*E(7)^2-2*E(7)^5+E(7)^6 C = -2*E(7)+E(7)^3+E(7)^4-2*E(7)^6 D = -E(4) = -Sqrt(-1) = -i