# Properties

 Label 14T51 Degree $14$ Order $21504$ Cyclic no Abelian no Solvable no Primitive no $p$-group no

# Related objects

## Group action invariants

 Degree $n$: $14$ Transitive number $t$: $51$ CHM label: $[2^{7}]L(7)=2wrL(7)$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $2$ Generators: (1,9,11)(2,4,8)(3,13,5)(6,12,10), (2,4)(5,13)(6,12)(9,11), (7,14), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$168$:  $\GL(3,2)$
$336$:  14T17
$1344$:  $C_2^3:\GL(3,2)$
$2688$:  14T43
$10752$:  14T50

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

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

## Low degree siblings

14T51, 28T462 x 2, 28T463, 28T464 x 2, 42T767 x 2, 42T768 x 2, 42T769 x 2, 42T770 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $7$ $2$ $( 7,14)$ $2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $21$ $2$ $( 2, 9)( 7,14)$ $2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $28$ $2$ $( 2, 9)( 4,11)( 7,14)$ $2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $7$ $2$ $( 2, 9)( 6,13)( 7,14)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $28$ $2$ $( 2, 9)( 4,11)( 6,13)( 7,14)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $7$ $2$ $( 2, 9)( 4,11)( 5,12)( 6,13)$ $2, 2, 2, 2, 2, 1, 1, 1, 1$ $21$ $2$ $( 2, 9)( 4,11)( 5,12)( 6,13)( 7,14)$ $2, 2, 2, 2, 2, 2, 1, 1$ $7$ $2$ $( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$ $2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $84$ $2$ $( 3, 5)( 6, 7)(10,12)(13,14)$ $4, 2, 2, 1, 1, 1, 1, 1, 1$ $168$ $4$ $( 3, 5)( 6,14,13, 7)(10,12)$ $2, 2, 2, 2, 2, 1, 1, 1, 1$ $84$ $2$ $( 2, 9)( 3, 5)( 6, 7)(10,12)(13,14)$ $4, 2, 2, 2, 1, 1, 1, 1$ $168$ $4$ $( 2, 9)( 3, 5)( 6,14,13, 7)(10,12)$ $2, 2, 2, 2, 2, 1, 1, 1, 1$ $168$ $2$ $( 3, 5)( 4,11)( 6, 7)(10,12)(13,14)$ $4, 2, 2, 2, 1, 1, 1, 1$ $336$ $4$ $( 3, 5)( 4,11)( 6,14,13, 7)(10,12)$ $2, 2, 2, 2, 2, 2, 1, 1$ $168$ $2$ $( 2, 9)( 3, 5)( 4,11)( 6, 7)(10,12)(13,14)$ $4, 2, 2, 2, 2, 1, 1$ $336$ $4$ $( 2, 9)( 3, 5)( 4,11)( 6,14,13, 7)(10,12)$ $4, 4, 1, 1, 1, 1, 1, 1$ $84$ $4$ $( 3,12,10, 5)( 6,14,13, 7)$ $4, 4, 2, 1, 1, 1, 1$ $84$ $4$ $( 2, 9)( 3,12,10, 5)( 6,14,13, 7)$ $4, 4, 2, 1, 1, 1, 1$ $168$ $4$ $( 3,12,10, 5)( 4,11)( 6,14,13, 7)$ $4, 4, 2, 2, 1, 1$ $168$ $4$ $( 2, 9)( 3,12,10, 5)( 4,11)( 6,14,13, 7)$ $2, 2, 2, 2, 2, 2, 1, 1$ $84$ $2$ $( 1, 8)( 3, 5)( 4,11)( 6, 7)(10,12)(13,14)$ $4, 2, 2, 2, 2, 1, 1$ $168$ $4$ $( 1, 8)( 3, 5)( 4,11)( 6,14,13, 7)(10,12)$ $2, 2, 2, 2, 2, 2, 2$ $84$ $2$ $( 1, 8)( 2, 9)( 3, 5)( 4,11)( 6, 7)(10,12)(13,14)$ $4, 2, 2, 2, 2, 2$ $168$ $4$ $( 1, 8)( 2, 9)( 3, 5)( 4,11)( 6,14,13, 7)(10,12)$ $4, 4, 2, 2, 1, 1$ $84$ $4$ $( 1, 8)( 3,12,10, 5)( 4,11)( 6,14,13, 7)$ $4, 4, 2, 2, 2$ $84$ $4$ $( 1, 8)( 2, 9)( 3,12,10, 5)( 4,11)( 6,14,13, 7)$ $4, 4, 2, 2, 1, 1$ $672$ $4$ $( 2, 3, 4, 7)( 5, 6)( 9,10,11,14)(12,13)$ $8, 2, 2, 1, 1$ $672$ $8$ $( 2, 3, 4,14, 9,10,11, 7)( 5, 6)(12,13)$ $4, 4, 4, 1, 1$ $672$ $4$ $( 2, 3, 4, 7)( 5,13,12, 6)( 9,10,11,14)$ $8, 4, 1, 1$ $672$ $8$ $( 2, 3, 4,14, 9,10,11, 7)( 5,13,12, 6)$ $4, 4, 2, 2, 2$ $672$ $4$ $( 1, 8)( 2, 3, 4, 7)( 5, 6)( 9,10,11,14)(12,13)$ $8, 2, 2, 2$ $672$ $8$ $( 1, 8)( 2, 3, 4,14, 9,10,11, 7)( 5, 6)(12,13)$ $4, 4, 4, 2$ $672$ $4$ $( 1, 8)( 2, 3, 4, 7)( 5,13,12, 6)( 9,10,11,14)$ $8, 4, 2$ $672$ $8$ $( 1, 8)( 2, 3, 4,14, 9,10,11, 7)( 5,13,12, 6)$ $3, 3, 3, 3, 1, 1$ $896$ $3$ $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)$ $6, 3, 3, 1, 1$ $896$ $6$ $( 2, 3, 5)( 4,14,13,11, 7, 6)( 9,10,12)$ $6, 3, 3, 1, 1$ $896$ $6$ $( 2, 3, 5, 9,10,12)( 4, 7, 6)(11,14,13)$ $6, 6, 1, 1$ $896$ $6$ $( 2, 3, 5, 9,10,12)( 4,14,13,11, 7, 6)$ $3, 3, 3, 3, 2$ $896$ $6$ $( 1, 8)( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)$ $6, 3, 3, 2$ $896$ $6$ $( 1, 8)( 2, 3, 5)( 4,14,13,11, 7, 6)( 9,10,12)$ $6, 3, 3, 2$ $896$ $6$ $( 1, 8)( 2, 3, 5, 9,10,12)( 4, 7, 6)(11,14,13)$ $6, 6, 2$ $896$ $6$ $( 1, 8)( 2, 3, 5, 9,10,12)( 4,14,13,11, 7, 6)$ $7, 7$ $1536$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)$ $14$ $1536$ $14$ $( 1, 2, 3, 4, 5, 6,14, 8, 9,10,11,12,13, 7)$ $7, 7$ $1536$ $7$ $( 1, 2, 3, 7, 6, 4, 5)( 8, 9,10,14,13,11,12)$ $14$ $1536$ $14$ $( 1, 2, 3,14,13,11,12, 8, 9,10, 7, 6, 4, 5)$

## Group invariants

 Order: $21504=2^{10} \cdot 3 \cdot 7$ Cyclic: no Abelian: no Solvable: no GAP id: not available
 Character table: not available.