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Properties

Label 14T51
Order \(21504\)
n \(14\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

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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)
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)
$|\Aut(F/K)|$:  $2$

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 TypeSizeOrderRepresentative
$ 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:  Data not available
Character table: Data not available.