Show commands:
Magma
magma: G := TransitiveGroup(14, 19);
Group action invariants
Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $19$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $\GL(3,2) \times C_2$ | ||
CHM label: | $L(7)[x]2$ | ||
Parity: | $-1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | (1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (2,4)(5,13)(6,12)(9,11), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $168$: $\GL(3,2)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 7: $\GL(3,2)$
Low degree siblings
14T17 x 2, 14T19, 16T714, 28T43 x 2, 42T78, 42T79, 42T80 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
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 $ | $21$ | $2$ | $( 3, 5)( 6,14)( 7,13)(10,12)$ |
$ 4, 4, 2, 2, 1, 1 $ | $42$ | $4$ | $( 2, 4)( 3, 5, 7,13)( 6,10,12,14)( 9,11)$ |
$ 3, 3, 3, 3, 1, 1 $ | $56$ | $3$ | $( 2, 6,10)( 3, 9,13)( 4,12,14)( 5, 7,11)$ |
$ 2, 2, 2, 2, 2, 2, 2 $ | $21$ | $2$ | $( 1, 2)( 3, 6)( 4,11)( 5,12)( 7,14)( 8, 9)(10,13)$ |
$ 4, 4, 2, 2, 2 $ | $42$ | $4$ | $( 1, 2)( 3,12,13,14)( 4,11)( 5, 6, 7,10)( 8, 9)$ |
$ 14 $ | $24$ | $14$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$ |
$ 6, 6, 2 $ | $56$ | $6$ | $( 1, 2, 3, 8, 9,10)( 4, 5,14,11,12, 7)( 6,13)$ |
$ 14 $ | $24$ | $14$ | $( 1, 2, 3,14,13, 4, 5, 8, 9,10, 7, 6,11,12)$ |
$ 7, 7 $ | $24$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$ |
$ 7, 7 $ | $24$ | $7$ | $( 1, 3, 9,11,13, 7, 5)( 2, 4, 6,14,12, 8,10)$ |
$ 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $336=2^{4} \cdot 3 \cdot 7$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | no | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 336.209 | magma: IdentifyGroup(G);
|
Character table: |
2 4 4 3 1 4 3 1 1 1 1 1 4 3 1 . . 1 . . . 1 . . . 1 7 1 . . . . . 1 . 1 1 1 1 1a 2a 4a 3a 2b 4b 14a 6a 14b 7a 7b 2c 2P 1a 1a 2a 3a 1a 2a 7a 3a 7b 7a 7b 1a 3P 1a 2a 4a 1a 2b 4b 14b 2c 14a 7b 7a 2c 5P 1a 2a 4a 3a 2b 4b 14b 6a 14a 7b 7a 2c 7P 1a 2a 4a 3a 2b 4b 2c 6a 2c 1a 1a 2c 11P 1a 2a 4a 3a 2b 4b 14a 6a 14b 7a 7b 2c 13P 1a 2a 4a 3a 2b 4b 14b 6a 14a 7b 7a 2c X.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 X.3 3 -1 1 . -1 1 A . /A A /A 3 X.4 3 -1 1 . -1 1 /A . A /A A 3 X.5 3 -1 1 . 1 -1 -/A . -A /A A -3 X.6 3 -1 1 . 1 -1 -A . -/A A /A -3 X.7 6 2 . . 2 . -1 . -1 -1 -1 6 X.8 6 2 . . -2 . 1 . 1 -1 -1 -6 X.9 7 -1 -1 1 -1 -1 . 1 . . . 7 X.10 7 -1 -1 1 1 1 . -1 . . . -7 X.11 8 . . -1 . . 1 -1 1 1 1 8 X.12 8 . . -1 . . -1 1 -1 1 1 -8 A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 |
magma: CharacterTable(G);